Over view of Real Analysis (Eagle View of Real Analysis)

Contents

1 Analysis
1.1 Set Theory
1.2 Functions
1.3 Sequence
1.4 Set Algebra
1.5 Countable sets
1.6 Relations
1.6.1 Partial Orderings
1.7 The Real Number Axioms
1.8 Integers
1.9 Integers, Rational Numbers as Subset of $ℝ$
1.10 Extended Real Number System
1.11 Sequence of Real Numbers
1.12 Limit Superior and Limit Inferior
1.13 Sequences
1.14 Infinite Series
1.15 Limits, Continuity and Uniform Continuity of Function
1.16 Differentiability of Function
1.17 Sequence and Series of Function
1.18 Riemann (Stieltjes) Integrals
1.19 Improper Riemann Integral
1.20 Types of Improper Integrals
1.21 Uniform Convergence
1.22 Discontinuities of Real Valued Function
1.23 Monotonic Functions
1.24 Functions of Bounded Variation
1.25 Absolutely Continuous Function
1.26 Elements of Metric Spaces
1.27 Lebesgue Measure
1.27.1 Measure
1.27.2 Lebesgue Outer Measure
1.27.3 Lebesgue Measurable Sets and Lebesgue Measure
1.27.4 Lebesgue Measurable Functions
1.27.5 Littlewood’s Three Principles
1.28 The Lebesgue Integration
1.28.1 The Riemann Integral
1.28.2 Step function
1.28.3 The Lebesgue Integral of a Bounded Function Over a Set of Finite Measure
1.28.4 The Integral of a Non-negative Function
1.28.5 The General Lebesgue Integral
1.28.6 Convergence in Measure
1.29 Real Valued Functions of Several Variables
1.30 Partial Derivatives
1.31 Implicit Functions and Inverse Functions
1.32 Extrema for Real Valued Functions
Index

Analysis

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1.1 Set Theory

1. Set: A collection of elements with some property $X=\{x:P(x\left)\text{is property}\right\}$.

2. Subset: $A\subset X\Rightarrow \forall <!--FUNCTION\; APPLICATION-->x\in A,x\in X$.

3. Complement: $A\subset X{A}^c=\{x:x\notin A\text{And}x\in X\}$.

4. Union: $A\cup B=\{x:x\in A\text{or}x\in B\}$.

5. Intersection: $A\cap B=\{x:x\in A\text{And}x\in B\}$.

6. Difference: $B\sim A=\{x:x\in B\text{And}x\notin A\}$.

7. Symmetric Difference: $A\mathrm{\Delta }B=(A\sim B)\cup (B\sim A)$.

8. Disjoint Sets: $A\cap B=\varnothing$.

9. De-Morgan’s law:

   $$\begin{array}{llll}\hfill {(A\cup B)}^c& =A^c\cap B^c.& \hfill & \\ \hfill {(A\cap B)}^c& =A^c\cup B^c.& \hfill & \end{array}$$

1.2 Functions

1. Function: A function $f$ from $X$ into set $Y$ is a rule that assign to each $x\in X$, there is $y=f\left(x\right)\in Y$.

2. Onto function: $f:X\to Y$ and $f\left(X\right)=Y$ then $f$ is onto..

3. Inverse function: $f:X\to Y$, then $f^{-1}\left(B\right)=\{x\in X:f(x)\in B\},B\subset Y.$

4. One-to-one function: (Univalent, injective) $f:X\to Y$, if $f\left(x_1\right)=f\left(x_2\right)\iff x_1=x_2$.

5. One-to-one correspondence: (Bijective) $f:X\to Y$. If $f$ is one-to-one and onto.

6. Composition: $f:X\to Y,g:Y\to Z$, then $h:X\to Z:h\left(x\right)=g\left(f\right(x\left)\right)$ or $g\circ f$.

7. Restriction: $f:X\to Y,A\subset X,g:A\to Y,g\left(x\right)=f\left(x\right)$ for $x\in A$. Then $g$ is the restriction of $f$ to $A$.

1.3 Sequence

1. Finite sequence: A function whose domain is the first $n$ natural number. i.e., $\{i\in \mathbb{N}:i\le n\}$.

2. Infinite sequence: A function whose domain is set of natural number.

3. Countable set: Set is countable if it is the range of some sequence.

4. Finite countable set: Set is finite countable if it is a range of some finite sequence.

5. Monotone function: $g:\mathbb{N}\to \mathbb{N}$ if $i>j\Rightarrow g\left(i\right)>g\left(j\right)$ or $g\left(i\right)<g\left(j\right)$.

1.4 Set Algebra

1. Algebras of sets (Boolean Algebra): A collection $\mathcal{𝒜}$ of subsets of $X$ is an algebra of sets if

   1. $A,B\in \mathcal{𝒜}\Rightarrow A\cup B\Rightarrow \mathcal{𝒜}$.

   2. $A\in \mathcal{𝒜}\Rightarrow A^c\in \mathcal{𝒜}$.

   3. $A,B\in \mathcal{𝒜}\Rightarrow A\cap B\Rightarrow \mathcal{𝒜}$.

2. $\sigma -$ algebra (Borel sets): An algebra $\mathcal{𝒜}$ is $\sigma -$ algebra if every union of countable collection of sets in $\mathcal{𝒜}$ is again in $\mathcal{𝒜}$.

3. Some Important Theorems

   1. Given any collection $𝒞$ of subsets of $X$, there is a smallest algebra $\mathcal{𝒜}$ which contains $𝒞$.

   2. If $\mathcal{𝒜}$ is an algebra of subsets and $\left\{A_i\right\}$ a sequence of sets in $\mathcal{𝒜}$. Then there is a sequence $\left\{B_i\right\}$ os sets in $\mathcal{𝒜}$ such that $B_n\cap B_m=\varnothing$ for $n\ne m$ and $\underset{i=1}{\overset{\infty }{\bigcup <!--FUNCTION\; APPLICATION-->}}{B}_i=\underset{i=1}{\overset{\infty }{\bigcup <!--FUNCTION\; APPLICATION-->}}{A}_i$.

1.5 Countable sets

1. Finite set: A set is called finite if it is either empty or the range of finite sequence.

2. Countable set A set is countable if it is either empty or the range of a sequence.

3. Some Important Theorems

   1. Every subset of countable set is countable.

   2. If $A$ is a countable set. Then the set of all finite sequences from $A$ is also countable.

   3. The set of all rational numbers is countable.

   4. The union of a countable collection of countable set is countable

1.6 Relations

1. Relations: $R$ is a relation of set $X$ if $xRy\Rightarrow x\in X$ and $y\in X$.

2. Reflexive relation: If $\forall <!--FUNCTION\; APPLICATION-->x\in X,xRx$.

3. Symmetric relation: If $xRy$ and $yRx,\forall <!--FUNCTION\; APPLICATION-->x,y\in X$.

4. Transitive relation: If $xRy$ and $yRz\Rightarrow xRz,\forall <!--FUNCTION\; APPLICATION-->x,y,z\in Z$.

5. Equivalence relation: A relation that is reflexive, symmetric and transitive on set $X$.

6. Anti symmetric relation: If $xRy$ and $yRx$ then $y=x$.

1.6.1 Partial Orderings

1. Partial ordering: $\prec$ is partial ordering on set $X$ if it transitive and antisymmetric.

2. Linear ordering: A linear ordering on a set $X$ if $x,y\in Xx\prec y$ or $y\prec x$.

3. $a$ proceeds $b$: $a,b\in X$ and $a\prec b$.

4. Minimal element: $a\in X$ is a minimal element of $E\subset X$ there is no $x\ne a\in E:x\prec a$.

5. Hausdorff maximal principle: Let $\prec$ be a partially ordering set $X$. Then there is a maximal linearly ordered subset $S$ of $X.$

1.7 The Real Number Axioms

Let $ℝ$ be the set of real numbers

1. Field Axioms: For all real numbers $x,y,z\in ℝ$.

   1. $x+y=y+x$ and $xy=yx$ (commutative law)

   2. $x+(y+z)=(x+y)+z$ and $x\left(yz\right)=\left(xy\right)z$ (associative law)

   3. $x(y+z)=xy+xz$ (distributive law)

   4. $\exists <!--FUNCTION\; APPLICATION-->0\in ℝ:x+0=x\forall <!--FUNCTION\; APPLICATION-->x\in ℝ$ Existence of additive identity ’0’

   5. $\exists <!--FUNCTION\; APPLICATION-->1\in ℝ:x.1=x\forall <!--FUNCTION\; APPLICATION-->x\in ℝ$ Existence of multiplicative identity ’1’

   6. $\forall <!--FUNCTION\; APPLICATION-->x\in ℝ,\exists <!--FUNCTION\; APPLICATION-->-x\in ℝ$ such that $x+(-x)=0$ (Existence of additive inverse)

   7. $\forall <!--FUNCTION\; APPLICATION-->x\in ℝ,\exists <!--FUNCTION\; APPLICATION-->x^{-1}\in ℝ:x{x}^{-1}=1$ Existence of multiplicative inverse $x^{-1}=\frac{1}{x}$

2. The order of axioms: For all real numbers $x,y,z\in ℝ$

   1. Exactly one relation holds $x=y,x<y,y<x$.

   2. If $x<y$, then for every $z,x+z<y+z$.

   3. If $x>0$ and $y>0$ then, $xy>0$.

   4. If $x>y$ and $y>z$ then $x>z$.

3. Positive real numbers: ${ℝ}^{+}=\{x\in ℝ:x>0\}$.
   Negative real numbers: ${ℝ}^{-}=\{x\in ℝ:x<0\}$.

4. Order relation: $x\le y\Rightarrow$ either $x<y$ or $x=y$. $x\ge y\Rightarrow$ either $x>y$ or $x=y$.

5. Ordered field: A field that satisfies ordered axioms

6. Intervals: open interval : $(a,b)=\{x:a<x<b\}$. Closed Interval : $[a,b]=\{x:a\le x\le b\}$. Half closed intervals:

   $$\begin{array}{llll}\hfill (a,b]& =\{x:a<x\le b\}& \hfill & \\ \hfill [a,b)& =\{x:a\le x<b\}.& \hfill & \end{array}$$

7. Bounds: Let $S\subset ℝ$ then

   1. Upper bound of $S$: $b\in ℝ$ is an upper bound of $S$ if $\forall <!--FUNCTION\; APPLICATION-->x\in S,x\le b$.

   2. Lower bound of $S$: $b\in ℝ$ is an lower bound of $S$ if $\forall <!--FUNCTION\; APPLICATION-->x\in S,x\ge b$.

   3. Maximal element of $S$: $b\in S$ is a maximal element if it is an upper bound of $S$.

   4. Minimal element of $S$: $b\in S$ is a maximal element if it is an lower bound of $S$.

   5. Least upper bound of $S$ ($\sup <!--FUNCTION\; APPLICATION-->S$): $c\in ℝ$ is a least upper bound of $S,\sup <!--FUNCTION\; APPLICATION-->S=c$ if

      1. $c$ is an upper bound of $S$,

      2. $c\le b$ for each upper bound $b$ of $S$.

   6. Greatest lower bound of $S$ ($\sup <!--FUNCTION\; APPLICATION-->S$): $c\in ℝ$ is a greatest lower bound of $S,\inf <!--FUNCTION\; APPLICATION-->S=c$ if

      1. $c$ is a lower bound of $S$,

      2. $c\ge b$ for each lower bound $b$ of $S$.

   7. Completeness axioms:

      1. Every non-empty set $S$ of real numbers which has an upper bound has least upper bound.

      2. Every non-empty set $S$ of real numbers which has an lower bound has greatest lower bound.

   8. Some Important Theorems

   1. Let $L$ and $U$ be non-empty subset of $ℝ$ with $ℝ=L\cup U$ and for each $l\in L$ and $u\in U$ we have $l<u$ then either $L$ has a greatest (maximal) or $U$ has a least (minimal) element.

   2. Approximation property:

      1. Let $S$ be a non-empty set of real numbers with $\sup <!--FUNCTION\; APPLICATION-->S=b$, then $\forall <!--FUNCTION\; APPLICATION-->a<b,\exists <!--FUNCTION\; APPLICATION-->x\in S:a<x\le b$.

      2. Let $S$ be a non-empty set of real numbers with $\inf <!--FUNCTION\; APPLICATION-->S=a$, then $\forall <!--FUNCTION\; APPLICATION-->a<b,\exists <!--FUNCTION\; APPLICATION-->x\in S:a\le x<b$.

   3. Additive property: Given non-empty subsets $A$ and $B$ of $ℝ$, let $C=\{x+y:x\in A\text{And}y\in B\}$.

      1. If $A$ and $B$ has supremum, then $\sup <!--FUNCTION\; APPLICATION-->C=\sup <!--FUNCTION\; APPLICATION-->A+\sup <!--FUNCTION\; APPLICATION-->B$.

      2. If $A$ and $B$ has infimum , then $\sup <!--FUNCTION\; APPLICATION-->C=\inf <!--FUNCTION\; APPLICATION-->A+\inf <!--FUNCTION\; APPLICATION-->B$.

   4. Comparison property: For non-empty subsets $S$ and $T$ of $ℝ$ such that $s\le t$ for every $s\le S$ and $t\in T$.

      1. If $T$ has a supremum then $S$ has supremum and $\sup <!--FUNCTION\; APPLICATION-->S\le \sup <!--FUNCTION\; APPLICATION-->T$.

      2. If $S$ has a minimum then $T$. has infimum and $\inf <!--FUNCTION\; APPLICATION-->S\le \inf <!--FUNCTION\; APPLICATION-->T$.

   5. $\inf <!--FUNCTION\; APPLICATION-->x=-\sup <!--FUNCTION\; APPLICATION-->(-x)$.

1.8 Integers

1. Inductive set: A set $S$ of real number is inductive set if it satisfies principle of induction.i.e.,

   1. $1\in S$,

   2. $\forall <!--FUNCTION\; APPLICATION-->x\in S,x+1\in S$.

2. Positive integer: A real number is a positive integer, if it belongs to every inductive set, ${\mathbb{Z}}^{+}=\{1,2,3,\dots <!--FUNCTION\; APPLICATION-->\}$.

3. Negative integer: A negative positive integer ${\mathbb{Z}}^{-}=\{-1,-2,-3,\dots <!--FUNCTION\; APPLICATION-->\}$.

4. Integers: Positive integers, negative integers and zero $\mathbb{Z}={\mathbb{Z}}^{+}\cup {\mathbb{Z}}^{-}\cup \left\{0\right\}$.

5. Divisor and multiple: $n$ and $d$ are integers if $n=cd$ for some $c$ an integer, then $d$ is a divisor of $n$ or $n$ is a multiple of $d$ denoted by $d|n$.

6. Prime: If $n$ is an integer, then $n$ is prime number if $n>1$ and positive divisor of $n$ are 1 and itself

7. Composite number: If $n>1$ and $n$ is not a prime then $n$ is composite number.

8. Some Important Theorems:

   1. Every integer $n>1$ is either prime or a product of primes.

   2. Every pair of integers $a$ and $b$ has a common divisor $d$ of the form $d=ax+by$, where $x,y$ are integers and every common divisor of $a$ and $b$ divides $d$.

   3. Euclid lemma: If $a|bc$ and $(a,b)=1$ then $a|c$.

   4. If a prime $p$ divides $ab$ then $p|a$ or $p|b$. If a prime $p$ divides a product $a_1,a_2,a_3.\dots <!--FUNCTION\; APPLICATION-->$ then $p$ divides atleast one of the factors $a_1,a_2,a_3,\dots <!--FUNCTION\; APPLICATION-->$

   5. Unique factorization theorem: Every integer $n>1$ can be represented as a product of prime factors in only way, apart from the order of the factors.

1.9 Integers, Rational Numbers as Subset of $ℝ$

Archimedes axiom: Given any real number $x$, there is an integer $n$ such that $x<n$.

1. Every ordered field contains the integers, the natural numbers and the rational numbers.

2. Between any two real numbers is a rational number, i,e., if $x<y$, there is a rational number $r:x<r<y$.

3. The set ${\mathbb{Z}}^{+}$ of positive integers $1,2,3\dots <!--FUNCTION\; APPLICATION-->$ is unbounded above.

1.10 Extended Real Number System

The extended real number system is the set of real numbers $ℝ$, together with two symbols $+\infty$ and $-\infty$ which satisfied.

1. If $x\in ℝ$, then

   $$\begin{array}{llllllll}\hfill x+(+\infty )& =+\infty ,x+(-\infty )=-\infty x-(+\infty )& \hfill =-\infty ,x-(-\infty )=+\infty \frac{x}{+\infty }& =\frac{x}{-\infty }=0.& \hfill & & \hfill & \end{array}$$

2. If $x>0$ then $x(+\infty )=+\infty ,x(-\infty )=-\infty$.

3. If $x<0$ then $x(+\infty )=-\infty ,x(-\infty )=+\infty$

4. $(+\infty )+(+\infty )=(+\infty )(+\infty )=(-\infty )(-\infty )=+\infty$.

5. $x\in ℝ$ then $-\infty <x+\infty$.

1. Important: $\infty -\infty$ is undefined, $\frac{0}{0},\frac{\infty }{\infty },\frac{-\infty }{-\infty }$ is also undefined and $0.\infty =0$.

2. Boundedness and extended real number system:

   1. The set $S\subset ℝ$ is bounded above if $\sup <!--FUNCTION\; APPLICATION-->S<+\infty$.

   2. The set $S\subset ℝ$ is bounded below if $\inf <!--FUNCTION\; APPLICATION-->S>-\infty$.

1.11 Sequence of Real Numbers

1. Sequence: $\left\{a_n\right\}$ of real number is a function whose domain is the set of natural numbers.

2. Limit: A real number $l$ is a limit of the sequence $\left\{a_n\right\},\lim <!--FUNCTION\; APPLICATION-->a_n=l$ if $𝜖>0,\exists <!--FUNCTION\; APPLICATION-->N\in \mathbb{N}$ such that $|a_n-l|<𝜖,\forall <!--FUNCTION\; APPLICATION-->n\ge N$.

3. Cauchy sequence: Sequence $\left\{a_n\right\}$ is Cauchy sequence if $𝜖>0,\exists <!--FUNCTION\; APPLICATION-->N$ such that $\forall <!--FUNCTION\; APPLICATION-->n\ge N$ and $m\ge N$ we have $|a_n-x_m|<𝜖$.

4. Convergent sequence: A sequence is called convergent sequence if it has a limit.

5. Divergent sequence: A sequence which is not a convergent sequence.

6. Cluster point: A real number $l$ is a cluster point of the sequence $\left\{a_n\right\}$ if for given $𝜖>0$, and given $N,n\ge N:|u_n-l|<𝜖.$

7. Some Important Results:

   1. A sequence of real number is convergent iff it is Cauchy sequence.

   2. If the limit of the sequence exist and it is unique.

   3. Every convergent sequence is bounded.

   4. If a sequence $\left\{a_n\right\}$ converges to $l$, then its subsequence also converges to $l.$

1.12 Limit Superior and Limit Inferior

1. Limit superior: Let $\left\{a_n\right\}$ is a sequence of real numbers. $U$ is called limit superior of $\left\{a_n\right\}$, $\mathrm{limsup}<!--FUNCTION\; APPLICATION-->a_n=\overline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n=\underset{n}{\inf <!--FUNCTION\; APPLICATION-->}\underset{k\ge n}{\sup <!--FUNCTION\; APPLICATION-->}{x}_k=U$ if

   1. given $𝜖>0,\exists <!--FUNCTION\; APPLICATION-->n:x_k<U+𝜖,\forall <!--FUNCTION\; APPLICATION-->k\ge n.$

   2. given $𝜖>0,$ and given $n:\exists <!--FUNCTION\; APPLICATION-->k\ge n$ such that $x_k>U-𝜖.$

2. Limit inferior: $\underline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n=\overline{\lim <!--FUNCTION\; APPLICATION-->}(-a_n)=\inf <!--FUNCTION\; APPLICATION-->\lim <!--FUNCTION\; APPLICATION-->a_n$.

3. Convergent sequence: A sequence for which $\overline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n=\underline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n$.

4. Some Important Results:

   1. $\underline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n\le \overline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n$

   2. $\underline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n=-\overline{\lim <!--FUNCTION\; APPLICATION-->}(-a_n)$

   3. $\begin{array}{ccccc}\hfill \underline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n+\underline{\lim <!--FUNCTION\; APPLICATION-->}{b}_n& =\underline{\lim <!--FUNCTION\; APPLICATION-->}(a_n+b_n)\hfill & \hfill =\underline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n+\overline{\lim <!--FUNCTION\; APPLICATION-->}{b}_n& =\overline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n+\overline{\lim <!--FUNCTION\; APPLICATION-->}{b}_n\hfill & \hfill =\overline{\lim <!--FUNCTION\; APPLICATION-->}(a_n+b_n)\end{array}$

   4. Sequence converges to $l$ iff $\underline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n=\overline{\lim <!--FUNCTION\; APPLICATION-->}{a}_n=\lim <!--FUNCTION\; APPLICATION-->a_n=l$

1.13 Sequences

1. Sequence: It is a function whose domain is the set of positive integers.

2. Examples:

   1. $\begin{array}{c}\hfill \{\frac{1}{n}\}=\{1,\frac{1}{2},\frac{1}{3},\dots <!--FUNCTION\; APPLICATION-->\}\end{array}$

   2. Let $p_n$ be the $n^{th}$ prime number, $\begin{array}{c}\hfill \left\{p_n\right\}=\{2,3,5,\dots <!--FUNCTION\; APPLICATION-->\}\end{array}$

3. Convergence of the sequence: A sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to a real number $A$ iff for each $𝜖>0$, there is a positive integer $N$, such that for all $n\ge N$, we have $|a_n-A|<𝜖.$

4. Neighbourhood: A set $N_x$ of real numbers is a neighbourhood of a real number $x$ iff $N_x$ contains an interval of positive length centred at $x$. i.e., iff there is $𝜖>0:(x-𝜖,x+𝜖)\subset N_x.$

5. Convergent and divergent sequence: A sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ is convergent iff there is a real number $A$ such that ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to $A$. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ is not convergent it is divergent sequence.

6. Cauchy sequence: A sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ is Cauchy sequence iff for each $𝜖>0$, there is a positive integer $N\text{such that if}m,n>N$ then $|a_n-a_m|<𝜖.$

7. Limit of a sequence: If a sequence is convergent, the unique number to which it converges is the limit of the sequence.

8. Accumulation point: For a set $S$ of real numbers, a real number $A$ is an accumulation point of $S$ iff every neighbourhood of $A$ contains infinitely many points of $S.$

9. Subsequence: Let ${\left\{a_n\right\}}_{n=1}^{\infty }$ be a sequence and ${\left\{n_k\right\}}_{k=1}^{\infty }$ be any sequence of positive integers such that $n_1<n_2<n_3<\dots <!--FUNCTION\; APPLICATION-->$. The sequence ${\left\{x_{n_k}\right\}}_{k=1}^{\infty }$ is called a subsequence of ${\left\{a_n\right\}}_{n=1}^{\infty }.$

10. Increasing sequence: Sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ is increasing, iff $a_n\le x_{n+1}$ for all $n.$

11. Decreasing sequence: Sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ is decreasing, iff $a_n\ge x_{n+1}$ for all $n.$

12. Monotone sequence: Sequence that is either increasing or decreasing.

13. Bounded above sequence: Sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ is bounded above, iff there exist a real number $N$ such that $a_n\le N$ for all $n.$

14. Bounded below sequence: Sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ is bounded above, iff there exist a real number $M$ such that $a_n\ge M$ for all $n.$

15. Bounded sequence: Sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ is bounded if it is bounded both from above and below $\iff$ there exist a real number $S$ such that $\left|a_n\right|\le S$ for all $n.$

16. Some Important Theorems:

    1. A sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to $A$ iff each neighbourhood of $A$ contains all but a finite number of terms of the sequence.

    2. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to real number $A$ and $B$, then $A=B.$

    3. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to $A$, then ${\left\{a_n\right\}}_{n=1}^{\infty }$ is bounded.

    4. Every convergent sequence is a Cauchy sequence.

    5. Every Cauchy sequence is bounded.

    6. Every Cauchy sequence is convergent.

    7. A sequence is Cauchy iff it is convergent.

    8. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to $A$ and ${\left\{b_n\right\}}_{n=1}^{\infty }$ converges to $B$, then

       1. ${\{a_n\pm b_n\}}_{n=1}^{\infty }$ converges to $A\pm B.$

       2. ${\left\{a_n{b}_n\right\}}_{n=1}^{\infty }$ converges to $AB.$

    9. If ${\left\{b_n\right\}}_{n=1}^{\infty }$ converges to $B$ and $b_n\ne 0$ for all $n$, then there exist a real number $M>0$ such that $\left|b_n\right|\ge M$ for all $n.$

    10. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to $A$ and ${\left\{b_n\right\}}_{n=1}^{\infty }$ converges to $B$, with $B\ne 0$ and $b_n\ne 0$ for all $n$, then $\{a_n∕b_n\}$ converges to $A∕B.$

    11. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to $A$ and ${\left\{b_n\right\}}_{n=1}^{\infty }$ converges to $B$, with $a_n\le b_n$ for all $n$, then $A\le B.$

    12. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to 0 and ${\left\{b_n\right\}}_{n=1}^{\infty }$ is bounded, then ${\left\{a_n{b}_n\right\}}_{n=1}^{\infty }$ converges to 0.

    13. A sequence converges iff each of its subsequence converges.

    14. A monotone sequence is convergent iff it is bounded.

    15. (Bolzano-Weierstrass theorem:) Every bounded infinite set of real numbers has atleast one accumulation point.

    16. Let $E$ be a set of real numbers. Then $x_0$ is an accumulation point of $E$, iff there is a sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ of numbers of $E$, each distinct from $x_0$ such that ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to $x_0.$

17. Some Important Results:

    1. Let ${\left\{a_n\right\}}_{n=1}^{\infty }$ and Let ${\left\{b_n\right\}}_{n=1}^{\infty }$ be tow sequences that converges to $A$. If ${\left\{c_n\right\}}_{n=1}^{\infty }$ is a sequence such that $a_n\le c_n\le b_n$ for all $n,$ then ${\left\{c_n\right\}}_{n=1}^{\infty }$ converges to $A.$

    2. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ and ${\left\{b_n\right\}}_{n=1}^{\infty }$ are Cauchy sequences, then ${\{a_n\pm b_n\}}_{n=1}^{\infty }$ is also a Cauchy sequence.

    3. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ and ${\left\{b_n\right\}}_{n=1}^{\infty }$ are two sequences. Also ${\left\{a_n\right\}}_{n=1}^{\infty }$ and ${\{a_n\pm b_n\}}_{n=1}^{\infty }$ are convergent, then ${\left\{b_n\right\}}_{n=1}^{\infty }$ is also convergent.

    4. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to $A$, then ${\left\{\right|a_n\left|\right\}}_{n=1}^{\infty }$ converges to $\left|A\right|.$

    5. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ is decreasing sequence (increasing) and bounded, then ${\left\{a_n\right\}}_{n=1}^{\infty }$ is convergent.

    1.14 Infinite Series

    1. Infinite series: An infinite series is a pair $({\left\{a_n\right\}}_{n=1}^{\infty },{\left\{S_n\right\}}_{n=1}^{\infty })$, where ${\left\{a_n\right\}}_{n=1}^{\infty }$ is a sequence of real numbers and $S_n=\underset{k=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{x}_k$ for all $n,a_n$ is the $n^{th}$ therm of the series and $S_n$ is the $n^{th}$ partial sum of the series.

    2. Convergence of series: If $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges, then ${\left\{S_n\right\}}_{n=1}^{\infty }$ converges.

    3. Converges absolutely: An infinite series $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges absolutely iff $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}\left|a_n\right|$ converges.

    4. Converges conditionally: An infinite series $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges absolutely but $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}\left|a_n\right|$ diverges.

    5. Cauchy’s product: Let $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ and $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n$ are two infinite series and for each $n$ define $c_n=\underset{k=0}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{y}_{n-k}$. The infinite series $\underset{k=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{c}_n$ is called the Cauchy’s product of two series $\underset{k=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ and $\underset{k=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n$.

    6. Rearrangement of series: Let $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ be an infinite series. If $T$ is one-ont function from $\{0,1,2,\dots <!--FUNCTION\; APPLICATION-->\}$ onto $\{0,1,2,\dots <!--FUNCTION\; APPLICATION-->\}$, then the infinite series $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{x}_{T_{\left(n\right)}}$ is called a rearrangement of $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$.

    7. Power series: Let ${\left\{a_n\right\}}_{n=0}^{\infty }$ be a sequence of real numbers. For each real number $x$, a series $\underset{k=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ is a power series.

    8. Interval of convergence: If $\underset{k=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ is a power series, then the set of points at which series converges is either

       1. $ℝ$, a set of all real numbers $(-\infty ,\infty )$ (Interval of finite radius)

       2. $\left\{0\right\}$ (Interval of zero radius)

       3. An interval of positive length centred at zero which may contain all ,none or one of its end points. These intervals are called interval of convergence.

    9. Radius of convergence: If $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ has an interval of convergence $C$ which is different from $ℝ$ and $\left\{0\right\}$, then there is a unique real number $r$ such that $(r,-r)\subset C\subset [r,-r]$. This number $r$ is called the radius of convergence of the power series.

18. Some Important Theorems:

    1. An infinite series $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges iff for each $𝜖>0$, there is $N$ such that $n\ge N$ and $p\ge 0$, then $|a_n+a_{n+1}+\cdots +a_{n+p}|<𝜖$.

    2. If $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges, then ${\left\{a_n\right\}}_{n=1}^{\infty }$ converges to zero.

    3. If $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ is absolutely convergent, then $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ is convergent.

    4. Comparison Test: Suppose $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ and $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n$ are infinite series with $b_n\ge 0$ for all $n$, then

       1. If $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n$ converges and there is $N_0$ such that $n\ge N_0\Rightarrow \left|a_n\right|\le b_n$, then $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges absolutely.

       2. If $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n$ diverges and there is $N_0$ such that $n\ge N_0\Rightarrow b_n\le \left|a_n\right|$, then $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ diverges.

    5. If $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ and $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n$ converges, then for real numbers $\alpha ,\beta ,\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}(\alpha a_n+\beta b_n)$ converges and $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}(\alpha a_n+\beta b_n)=\alpha \underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n+\beta \underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n.$

    6. If ${\left\{a_n\right\}}_{n=1}^{\infty }$ is a sequence of non-negative terms such that $a_n\ge a_{n+1}$ for all $n$, then $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges iff $\underset{k=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{2}^k{a}_{2^k}$ converges.

    7. The series $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}\frac{1}{n^p}$ converges iff $p>1.$

    8. Let $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ be an infinite series of non-zero terms. Then

       1. For real numbers $p\in (0,1)$ and positive integer $N$ such that $n\ge N\Rightarrow |\frac{a_{n+1}}{a_n}|\le p$, then $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges absolutely.

       2. If there is $N$ such that $n\ge N\Rightarrow |\frac{a_{n+1}}{a_n}|\ge 1$, then $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ diverges.

    9. Ratio test: If $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ is an infinite series of non-zero terms such that $\{a_{n+1}∕a_n{\}}_{n=1}^{\infty }$ converges to $L$. Then

       1. If $L<1$, the series converges absolutely.

       2. If $L>1$, the series diverges.

       3. If $L=1$, no conclusion concerning convergence can be made (Test Fail).

    10. Root test: If $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ is an infinite series. Then

        1. For real numbers $q,0\le q<1$ and $N$ such that $\sqrt[n]{\left|a_n\right|}\le$ for $n\ge N$, then $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges absolutely.

        2. If for infinitely many $n,\sqrt[n]{\left|a_n\right|}\ge 1$, then $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ diverges.

    11. If ${\left\{a_n\right\}}_{n=0}^{\infty }$ and ${\left\{b_n\right\}}_{n=0}^{\infty }$ are two sequences of real numbers, then

        1. Define $A_n=\underset{k=0}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{a}_k$ and $A_{-1}=0$. Then if $0\le p\le q,\begin{array}{ccc}\hfill \underset{n=p}{\overset{q}{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{b}_n& =\underset{n=p}{\overset{q-1}{\sum <!--FUNCTION\; APPLICATION-->}}{A}_n(b_n-b_{n+1})\hfill & \hfill +A_q{b}_q-A_{p-1}{b}_p.\end{array}$

        2. 1. Partial sums of $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ are bounded.

           2. $b_0\ge b_1\ge b_2\ge \dots <!--FUNCTION\; APPLICATION-->$ and

           3. ${\left\{b_n\right\}}_{n=0}^{\infty }$ converges to zero.

           Then $\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{b}_n$ converges.

    12. If ${\left\{b_n\right\}}_{n=0}^{\infty }$ is a sequence of real numbers such that

        1. Sequence ${\left\{b_n\right\}}_{n=0}^{\infty }$ converges to zero.

        2. $b_0\ge b_1\ge b_2\ge \cdots$

        Then $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{(-1)}^n{b}_n$ converges.

    13. If $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges absolutely and $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n$ converges with $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n=A$ and $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n=B$. Then Cauchy product $c_n=\underset{k=0}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{a}_k{b}_{n-k},\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{c}_n$ converges to $AB.$

    14. If $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges to $A$ and $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n$ converges to $B,$ and Cauchy product is $c_n=\underset{k=0}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{a}_k{b}_{n-k}$. If $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{c}_n$ converges to $C$ then $C=AB$

    15. If $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ is absolute convergent series converging to $A$ and $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_{T_n}$ is any arrangement of $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$. Then $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_{T_n}$ converges to $A.$

    16. Let $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ be a power series which converges for $x=x_0$ and diverges for $x=x_1$, then

        1. Power series $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ converges absolutely for $\left|x\right|<\left|x_0\right|.$

        2. Power series $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ diverges for $\left|x\right|>\left|x_1\right|.$

    17. If a power series $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ with $a_n\ne 0$ for all $n$ such that ${\{a_{n+1}∕a_n\}}_{n=0}^{\infty }$ converges to $L.$ Then

        1. If $L=0$, the series converges for all $x$.

        2. If $L\ne 0,1∕L$ is the radius of convergence.

    18. A power series $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$, with $a_n\ne 0$ for all $n$,

        1. Converges absolutely, if for any real number $q\ne 0$ and $N$ such that $n\ge N\Rightarrow |a_{n+1}∕a_n|\le q$.

        2. Diverges for $\left|x\right|\ge 1∕p$, if for real number $p>0$ and $N$ such that for all $n\ge N,|a_{n+1}∕a_n|\ge p$.

    19. A power series $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$,

        1. Converges for $x=0$, if sequence $\{\sqrt[n]{\left|a_n\right|}{\}}_{n=0}^{\infty }$ is unbounded.

        2. Converges for all $x$, if sequence $\{\sqrt[n]{\left|a_n\right|}{\}}_{n=0}^{\infty }$ converges to zero.

        3. Has radius of convergence $1∕a$, if $\{\sqrt[n]{\left|a_n\right|}{\}}_{n=0}^{\infty }$ is bounded and $a=\underset{n\to \infty }{\mathrm{limsup}<!--FUNCTION\; APPLICATION-->}\sqrt[n]{\left|a_n\right|}\ne 0.$

    20. An infinite series $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ diverges if $\{\sqrt[n]{\left|a_n\right|}{\}}_{n=0}^{\infty }$ is unbounded.

    21. If $\{\sqrt[n]{\left|a_n\right|}{\}}_{n=0}^{\infty }$ is bounded

        1. Then infinite series $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ converges absolutely, if $\underset{n\to \infty }{\mathrm{limsup}<!--FUNCTION\; APPLICATION-->}\sqrt[n]{\left|a_n\right|}<1$

        2. Then infinite series $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n$ diverges, if $\underset{n\to \infty }{\mathrm{limsup}<!--FUNCTION\; APPLICATION-->}\sqrt[n]{\left|a_n\right|}>1$

1.15 Limits, Continuity and Uniform Continuity of Function

1. Limits Let $f:D\to ℝ$ with $x_0$ an accumulation point of $D$. Then $f$ has a limit at $x_0$, defined by $\underset{x\to x_0}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)=L$ iff for every $𝜖>0$ there exist $\delta >0$ such that $0<|x-x_0|<\delta$ and $x\in D,\left|f\right(x)-L|<𝜖.$ Let $f:D\to ℝ$ and for all $x,y\in D,x\le y$

   1. $f$ is increasing function if $f\left(x\right)\le f\left(y\right)$.

   2. $f$ is decreasing function if $f\left(x\right)\ge f\left(y\right)$.

   3. $f$ is monotone function if it is either increasing or decreasing.

2. Some Important Theorems

   1. Let $f:D\to ℝ$ with $x_0$ an accumulation point of $D$, then $\underset{x\to x_0}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)$ exist

      1. If for each sequence ${\left\{x_n\right\}}_{n=1}^{\infty }$ converges to $x_0$, with $x_n\in D$ and $x_n\ne x_0$ for all $n,$ the sequence ${\left\{f\right(x_n\left)\right\}}_{n=1}^{\infty }$ converges.

      2. If for each sequence ${\left\{x_n\right\}}_{n=1}^{\infty }$ converges to $x_0$, with $x_n\in D∕\left\{x_0\right\}$ for all $n,$ the sequence ${\left\{f\right(x_n\left)\right\}}_{n=1}^{\infty }$ Cauchy.

      3. There is a neighbourhood $N_{x_0}$ of $x_0$ and a real number $M$ such that for all $x\in N_{x_0}\cap D,\left|f\right(x\left)\right|\le M.$

      4. If for each $𝜖>0$, there is a neighbourhood $N_{x_0}$ of $x_0$ such that $x,y\in N_{x_0}\cap D,x\ne x_0$ and $y\ne x_0$, we have $\left|f\right(x)-f(y\left)\right|<𝜖.$

   2. Let $f,g:D\to ℝ$ with $x_0$ an accumulation point of $D$ and $f,g$ have limits at $x_0$, then

      1. 1. $f\pm g$ has a limits at $x_0$.

         2. $fg$ has a limit at $x_0.$

         3. If $g\left(x\right)\ne 0$ for all $x\in D$ and $\underset{x\to x_0}{\lim <!--FUNCTION\; APPLICATION-->}g\left(x\right)\ne 0$ then $f∕g$ has limit at $x_0.$

      2. If $f\left(x\right)\le g\left(x\right)$ for all $x\in D$, then $\underset{x\to x_0}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)\le \underset{x\to x_0}{\lim <!--FUNCTION\; APPLICATION-->}g\left(x\right)$.

      3. If $f$ is bounded in a neighbourhood of $x_0$ and $\underset{x\to x_0}{\lim <!--FUNCTION\; APPLICATION-->}g\left(x\right)=0$, then $\underset{x\to x_0}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)g\left(x\right)=0$.

   3. If $f:D\to ℝ$ with $x_0$ an accumulation point of $D$. If $L_1$ and $L_2$ are the limits of $f$ at $x_0$, then $L_1=L_2$.

3. Continuity and Uniform Continuity

   1. Continuity: Let $E\subset ℝ,f:E\to ℝ$, if $x_0\in E$ then $f$ is continuous at $x_0$ iff for each $𝜖>0$ there exist $\delta >0$ such that if $|x-x_0|<\delta ,x\in E,\Rightarrow \left|f\right(x)-f(x_0\left)\right|<𝜖.$ If $f$ is continuous at $x$ for each $x\in E$, then $f$ is continuous (on $E$).

   2. Uniformly continuity: A function $f:D\to ℝ$ is uniformly continuous on $E\subset D$ iff for every $𝜖>0$, there is $\delta >0$ such that if $x,y\in E$ with $|x-y|<\delta$, then $\left|f{\left(x\right)}_f\right(y\left)\right|<𝜖$. If $f$ is uniformly continuous on $D$, $f$ is uniformly continuous.

   3. Closed set: A set $E\subset ℝ$ is closed iff every accumulation point of $E$ belongs to $E$.

   4. Open set: A set $A\subset ℝ$ is open iff for each $x\in A,$ there is a neighbourhood $N_x$ of $x$ such that $N_x\subset A$.

   5. Compact set: A set $E$ is compact iff for every family ${\left\{G_{\alpha }\right\}}_{\alpha \in A}$ of open sets $E\subset \underset{\alpha \in A}{\bigcup <!--FUNCTION\; APPLICATION-->}{G}_{\alpha }$, there is a finite set $\{{\alpha }_1,{\alpha }_2,\dots <!--FUNCTION\; APPLICATION-->,{\alpha }_n\}\subset A$ such that $E\subset \underset{i=1}{\overset{n}{\bigcup <!--FUNCTION\; APPLICATION-->}}{G}_{{\alpha }_i}$.

   6. Right continuous: If $E\subset ℝ,f;E\to ℝ$ and $x_0\in E$, the function $f$ is right continuous at $x_0$ iff for each $𝜖>0$, there is $\delta >0$ such that $x_0\le x\le x_0+\delta ,x\in E\Rightarrow \left|f\right(x)-f(x_0\left)\right|<𝜖.$

   7. Left continuous: If $E\subset ℝ,f;E\to ℝ$ and $x_0\in E$, the function $f$ is left continuous at $x_0$ iff for each $𝜖>0$, there is $\delta >0$ such that $x_0-\delta \le x\le x_0,x\in E\Rightarrow \left|f\right(x)-f(x_0\left)\right|<𝜖.$

4. Some Important Theorems

   1. Let $f:E\to ℝ$ with $x_0\in E$ and $x_0$ an accumulation point of $E$. Then these are equivalent

      1. The function $f$ is continuous at $x_0$.

      2. The function $f$ has a limit at $x_0$ and $\underset{x\to x_0}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)=f\left(x_0\right)$ exist.

      3. For every sequence $\left\{x_n\right\}$ converging to $x_0$ with $x_n\in E$ for each $n$, $\left\{f\right(x_n\left)\right\}$ converges to $f\left(x_0\right)$.

   2. If $f,g:D\to ℝ$ are continuous at $x_0\in D$. Then

      1. $f\pm g$ is continuous at $x_0$.

      2. $fg$ is continuous at $x_0$.

      3. $f∕g$ is continuous at $x_0$ if $g\left(x_0\right)\ne 0$.

   3. If $f:D\to ℝ,g:D^{\prime }\to ℝ$ with $f\left(D\right)\subset D^{\prime }$, where $f$ is continuous at $x_0\in D$ and $g$ is continuous at $f\left(x_0\right)$, then $g\circ f$ is continuous at $x_0.$

   4. If $f:D\to ℝ$, is continuous uniformly. Then if $x_0$ is an accumulation point of $D,f$ has a limit at $x_0$.

   5. A set $E\subset ℝ$ is closed iff $ℝ∕E$ is open.

   6. A set $E\subset ℝ$ is compact iff $E$ is closed and bounded.

   7. If $f:D\to ℝ$ is continuous with $D$ compact (i.e., closed and bounded). Then $f$ is uniformly continuous.

   8. If $f:E\to ℝ$ is continuous with $E$ compact, then $f\left(E\right)$ is compact.

   9. If $f:E\to ℝ$ is continuous and non-one with $E$ compact. Then $f^{-1}:f\left(E\right)\to E$ is continuous.

   10. Intermediate-value Theorem: If $f:[a,b]\to ℝ$ is continuous with $f\left(a\right)<y<f\left(b\right)$ (or) $f\left(b\right)<y<f\left(a\right)$, then there is $c\in (a,b)$ such that $f\left(c\right)=y$.

   11. If $f:[0,1]\to [0,1]$ is continuous, then there is $x\in [0,1]$ such that $f\left(x\right)=x.$

   12. If $f:[a,b]\to ℝ$ is continuous at and one-one, then $f$ is monotone.

   13. The set $[0,1]$ is uncountable.

1.16 Differentiability of Function

1. Differentiability: Let $f:D\to ℝ$, with $x_0$ is an accumulation point of $D$ and $x_0\in D$. For each $x\in D$, with $x\ne x_0$ define

   $$\begin{array}{lll}\hfill T\left(x\right)=\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}& & \hfill \end{array}$$

   The function $f$ is differentiable at $x_0$ (derivative at $x_0$ ) iff $T$ has a limit at $x_0$. i.e., $f^{\prime }\left(x_0\right)=\underset{x\to x_0}{\lim <!--FUNCTION\; APPLICATION-->}T\left(x\right)$ exist. The number $f^{\prime }\left(x_0\right)$ is derivative of $f$ at $x_0$. If $f$ is differentiable for each $x\in E\subset D$, then $f$ is differentiable on $E$.

2. Maximum (minimum) of function: Let $f:D\to ℝ$. A point $x_0\in D$ is relative maximum (minimum) of $f$ iff there is a neighbourhood $N_{x_0}$ of $x_0$ such that if $x\in N_{x_0}\cap D$, then $f\left(x\right)\le f\left(x_0\right)\left\{f\right(x)\ge f(x_0\left)\right\}$.

3. Some Important Theorems

   1. $f:D\to ℝ,x_0$ an accumulation point of $D$, then $f$ is differentiable at $x_0$ iff for every sequence ${\left\{x_n\right\}}_{n=1}^{\infty }$ for points of $D\left\{x_0\right\}$ converges to $x_0$, then the sequence $\{\frac{f\left(x_n\right)-f\left(x_0\right)}{x_n-x_0}{\}}_{n=1}^{\infty }$ converges.

   2. $f:D\to ℝ$ is differentiable at $x_0$. Then

      1. The point $x_0\in D$ and $x_0$ is an accumulation point of $D.$

      2. The function $f$ is continuous at $x_0$.

   3. If $f,g:D\to ℝ$ are differentiable at $x_0$, then

      1. ${(f\pm g)}^{\prime }\left(x_0\right)=f^{\prime }\left(x_0\right)\pm g^{\prime }\left(x_0\right)$ and $(f\pm g)$ is differentiable at $x_0$.

      2. $fg$ is differentiable at $x_0$, and ${\left(fg\right)}^{\prime }\left(x_0\right)=f^{\prime }\left(x_0\right)g\left(x\right)+f\left(x_0\right)g^{\prime }\left(x_0\right)$

      3. If $g\left(x_0\right)\ne 0$, then $f∕g$ (the domain is the set of all $x:g\left(x\right)\ne 0$) is differentiable at $x_0$ and $(\frac{f}{g}{)}^{\prime }\left(x_0\right)=\frac{f^{\prime }\left(x_0\right)g\left(x_0\right)-g^{\prime }\left(x_0\right)f\left(x_0\right)}{{\left[g\right(x_0\left)\right]}^2}$

   4. If $f:D\to ℝ$ and $g:D^{\prime }\to ℝ$ with $f\left(D\right)=D^{\prime }$. Then $f$ is differentiable at $x_0$ and $g$ is differentiable at $f\left(x_0\right)$. Also $g\circ f$ is differentiable at $x_0$ and ${(g\circ f)}^{\prime }=g^{\prime }\left\{f\right(x_0\left)\right\}{f}^{\prime }\left(x_0\right)$.

   5. If $n$ is an integer and $f\left(x\right)=x^n$, then $f$ is differentiable for all $x$ if $n>0$ and for all $x\ne 0$ if $n<0$ and $f^{\prime }\left(x\right)=n{x}^{n-1}.$ If $n=0$, then $f^{\prime }\left(x\right)=0$ for all $x.$

   6. If $f:[a,b]\to ℝ$ and $f$ has a relative minimum or a relative maximum at $x_0\in (a,b)$. If $f$ is differentiable at $x_0$, then $f^{\prime }\left(x_0\right)=0$.

   7. Rolle’s Theorem: If $f:[a,b]\to ℝ$ is continuous on $[a,b]$ and $f$ is differentiable on $(a,b)$. Then if $f\left(a\right)=f\left(b\right)=0$, there is $c\in (a,b)$ such that $f^{\prime }\left(c\right)=0$.

   8. Mean Value Theorem: If $f:[a,b]\to ℝ$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there is $c\in (a,b)$ such that $f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

   9. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Then

      1. If $f^{\prime }\left(x\right)\ne 0$ for all $x\in (a,b)$, then $f$ is one-one.

      2. If $f^{\prime }\left(x\right)=0$ for all $x\in (a,b)$, then $f$ is constant.

      3. If $f^{\prime }\left(x\right)>0$ for all $x\in (a,b)$, then $x<y$ and $x,y\in [a,b]\Rightarrow f\left(x\right)<f\left(y\right).$

      4. If $f^{\prime }\left(x\right)<0$ for all $x\in (a,b)$, then $x<y$ and $x,y\in [a,b]\Rightarrow f\left(x\right)>f\left(y\right).$

   10. If $f$ and $g$ are continuous on $[a,b]$ and differentiable on $(a,b)$ and that $f^{\prime }\left(x\right)=g^{\prime }\left(x\right)$ for all $x\in (a,b)$. Then there is a real number $\lambda$ such that $f\left(x\right)=g\left(x\right)+\lambda$ for all $x\in (a,b).$

   11. If $f$ is differentiable on $[a,b]$ and $\lambda$ is a real number such that $f^{\prime }\left(a\right)<\lambda <f^{\prime }\left(b\right)$ or $f^{\prime }\left(b\right)<\lambda <f^{\prime }\left(a\right)$. Then there is $c\in (a,b)$ such that $f^{\prime }\left(c\right)=\lambda .$

1.17 Sequence and Series of Function

1. Pointwise convergence: If ${\left\{f_n\right\}}_{n=0}^{\infty }$ is a sequence of functions and that $E$ is a subset of $ℝ$ such that $E\subset dom{f}_n$ for each integer $n>0$. Then ${\left\{f_n\right\}}_{n=0}^{\infty }$ converges pointwise on $E$ if for each $x\in E$, the sequence ${\left\{f_n\right(x\left)\right\}}_{n=0}^{\infty }$ converges. If ${\left\{f_n\right\}}_{n=0}^{\infty }$ converges pointwise on $E$, is defined by $f:E\to ℝ,f\left(x\right)=\underset{n\to \infty f_n\left(x\right)}{\lim <!--FUNCTION\; APPLICATION-->},$ for each $x\in E$.

2. Uniform convergence: A sequence ${\left\{f_n\right\}}_{n=1}^{\infty }$ of functions is said to converge uniformly of $E$ if there is a function $f:E\to ℝ$ such that for each $𝜖>0$, there is $N$ such that for each positive integer $n,n\ge N$ implies that $\left|f_n\right(x)-f(x\left)\right|<𝜖$, for each $x\in E.$

3. Some Important Theorems

   1. A sequence of functions ${\left\{f_n\right\}}_{n=1}^{\infty }$ converges uniformly on $E$ iff for each $𝜖>0$ there is a real number $N$ such that for all positive integer $m$ and $n,m\ge N$ and $n\ge N$ such that $\left|f_n\right(x)-f_m(x\left)\right|<𝜖$ for all $x\in E.$

   2. Weierstrass M-test: Suppose ${\left\{f_n\right\}}_{n=1}^{\infty }$ is a sequence of functions defined on $E$ and ${\left\{M_n\right\}}_{n=1}^{\infty }$ is a sequence of non-negative real numbers such that $\left|f_n\right(x\left)\right|\le M_n$ for $n=1,2,3,\dots <!--FUNCTION\; APPLICATION-->$ and for every $x\in E$. Then $\sum <!--FUNCTION\; APPLICATION-->f_n\left(x\right)$ converges uniformly on $E$ if $\sum <!--FUNCTION\; APPLICATION-->M_n$ converges.

   3. Suppose ${\left\{f_n\right\}}_{n=1}^{\infty }$ converges pointwise to $f$ on $E,x_0\in E,$ and infinitely many members of the sequence are continuous at $x_0$. If ${\left\{f_n\right\}}_{n=1}^{\infty }$ converges uniformly at $x_0$ then $f$ is continuous at $x_0.$

   4. If ${\left\{f_n\right\}}_{n=1}^{\infty }$ is a sequence of functions converging uniformly to $f$ on $E$ and for each positive integer $n,f_n$ is bounded on $E$, then $f$ is bounded on $E.$

   5. Let ${\left\{f_n\right\}}_{n=1}^{\infty }$ be a sequence of functions each is Riemann-integrable on $[a,b]$, converging uniformly to $f$ on $[a,b]$. Define $F_n\left(t\right)={\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}_a^{t}f\left(x\right)dx$ for each $t\in [a,b]$ and each $t\in [a,b]$ and each positive integer $n$. Then $f$ is Riemann-integrable on $[a,b]$, and ${\left\{F_n\right\}}_{n=1}^{\infty }$ converges uniformly on $[a,b]$ to function $F$ defined by $F\left(t\right)={\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}_a^{t}f\left(x\right)dx$ for each $t\in [a,b].$

   6. Suppose ${\left\{f_n\right\}}_{n=1}^{\infty }$ is a sequence of functions, each of which is differentiable on $[a,b]$. Suppose further that for some $x_0\in [a,b],{\left\{f_n\right(x_0\left)\right\}}_{n=1}^{\infty }$ converges and that ${\left\{f_n^{\prime }\right\}}_{n=1}^{\infty }$ converges uniformly to $g$ on $[a,b]$. Then

      1. ${\left\{f_n\right\}}_{n=1}^{\infty }$ converges uniformly on $[a,b]$ to a function $f$.

      2. $f$ is differentiable on $[a,b]$ and $f^{\prime }\left(x\right)=g\left(x\right)$ for all $x\in [a,b]$.

   7. Let $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ be a power series that converges for $-r<x<r,r>0$. Then $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ converges uniformly on $-t\le x\le t$ for each $0<t<r$.

   8. If ${\left\{b_n\right\}}_{n=1}^{\infty }$ is a bounded sequence of real numbers, then ${\left\{\sqrt[n]{n{b}_n}\right\}}_{n=1}^{\infty }$ is a bounded sequence of real numbers and $\underset{n\to \infty }{\mathrm{limsup}<!--FUNCTION\; APPLICATION-->}{b}_n=\underset{n\to \infty }{\mathrm{limsup}<!--FUNCTION\; APPLICATION-->}\sqrt[n]{n{b}_n}$.

   9. If $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ converges to $f$ on $(-r,r)$ with $r>0$. Then

      1. For each $0<t<r,\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n$ converges uniformly on $[-t,t]$.

      2. $f$ is $n-$times differentiable on $(-r,r)$ for each positive integer $n.$

      3. For each $0<t<r$ and each positive integer $m,$ $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}n(n-1)\cdots (n-m+1)a_n{x}^{n-m}$ converges uniformly on $[-t,t].$

      4. $f^m\left(0\right)=n!a_n$.

   10. Suppose ${\left\{a_n\right\}}_{n=0}^{\infty }$ and ${\left\{b_n\right\}}_{n=0}^{\infty }$ are two sequences of real numbers, $r>0$ and for all $x\in (-r,r)$, $\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{a}_n{x}^n=\underset{n=0}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}{b}_n{x}^n$. Then for each integer $n\ge 0,a_n=b_n.$

1.18 Riemann (Stieltjes) Integrals

1. Partition: A partition $P$ of $[a,b]$ is a finite set $\{x_0,x_1,\dots <!--FUNCTION\; APPLICATION-->,x_n\}$ such that $a=x_0<x_1<\dots <!--FUNCTION\; APPLICATION--><x_n=b$.

2. Refinement: If $P$ and $Q$ are partition of $[a,b]$ with $P\subset Q$, then $Q$ is a refinement of $P$

3. Riemann Integral: If $f:[a,b]\to ℝ$ is a bounded function and $P=\{x_1,x_2,\dots <!--FUNCTION\; APPLICATION-->,x_n\}$ is a partition of $[a,b]$. For each $i,(i=1,2,\dots <!--FUNCTION\; APPLICATION-->,n)$ define

   $$\begin{array}{lllllll}\hfill M_i\left(f\right)& =\sup <!--FUNCTION\; APPLICATION-->\left\{f\right(x):x\in [x_{i-1},x_i\left]\right\}{m}_i\left(f\right)& \hfill =\inf <!--FUNCTION\; APPLICATION-->\left\{f\right(x):x\in [x_{i-1},x_i\left]\right\}& & \hfill & & \hfill \\ \multicolumn{7}{c}{\text{Upper darbox sum of}f\text{,}}\\ \\ \hfill U(P,f)& =\sum _{i=1}^n{M}_i\left(f\right)(x_i-x_{i-1})& \hfill & & \hfill & \\ \multicolumn{6}{c}{\text{Lower darbox sum of}f\text{,}}\\ \\ \hfill L(P,f)& =\sum _{i=1}^n{m}_i\left(f\right)(x_i-x_{i-1})& \hfill & & \hfill & \end{array}$$
   $$\begin{array}{lllllllllllllll}\multicolumn{15}{c}{\text{Upper integral of}f\text{,}}\\ \hfill \underset{a}{\overset{-b}{\int \; <!--nolimits-->}}fdx& =\inf <!--FUNCTION\; APPLICATION-->\left\{U\right(P,f):P\text{is a partition}\}& \hfill & \\ \multicolumn{4}{c}{\text{Lower integral of}f\text{,}}\\ \\ \hfill \underset{-a}{\overset{b}{\int \; <!--nolimits-->}}fdx& =\sup <!--FUNCTION\; APPLICATION-->\left\{L\right(P,f):P\text{is a partition}\}& \hfill & \end{array}$$

4. Riemann Integrable on $[a,b]$: $f$ is Riemann integrable if $\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ exist. $\begin{array}{c}\hfill \underset{a}{\overset{-b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=\underset{-a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx\end{array}$

5. Riemann-Stielties integral: Suppose $f:[a,b]\to ℝ$ is bounded and $\alpha :[a,b]\to ℝ$ is an increasing function. For each partition $P=\{x_0,x_1,\dots <!--FUNCTION\; APPLICATION-->,x_n\}$ define
   $\begin{array}{ccc}\hfill U(P,f,\alpha )& =\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{M}_i\left(f\right)\left[x_i\right(\alpha )-x_{i-1}(\alpha \left)\right]\hfill & \hfill \\ \hfill L(P,f,\alpha )& =\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{m}_i\left(f\right)\left[x_i\right(\alpha )-x_{i-1}(\alpha \left)\right]\hfill \\ \hfill \underset{a}{\overset{-b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd\alpha & =\inf <!--FUNCTION\; APPLICATION-->\left\{U\right(P,f,\alpha ):P\text{is a partition}\}\hfill \\ \hfill \underset{-a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd\alpha & =\sup <!--FUNCTION\; APPLICATION-->\left\{L\right(P,f,\alpha ):P\text{is a partition}\}\hfill \end{array}$ $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a,b]$ if $\begin{array}{c}\hfill \underset{a}{\overset{-b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=\underset{-a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx\end{array}$ When $\alpha \left(x\right)=x$, the Riemann-Stieltijes integral with respect to $\alpha$ reduces to Riemann integral.

6. Selection of points: Let $P=\{x_0,x_1,\dots <!--FUNCTION\; APPLICATION-->,x_n\}$ be a partition of $[a,b]$. Then the collection of points $T=\{t_1,t_2,\dots <!--FUNCTION\; APPLICATION-->,t_n\}$ is called selection of points for $P$ if $x_{i-1}\le t_i\le x_i$, holds for $i=1,2,\dots <!--FUNCTION\; APPLICATION-->,n.$

7. Riemann sum associated with partition $P$:
   $\begin{array}{c}\hfill R_f(P,T)=\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}f\left(t_i\right)(x_i-x_{i-1})\end{array}$

8. Some Important Theorem

   1. Let $f:[a,b]\to ℝ$ be bounded and $\alpha :[a,b]\to ℝ$ be increasing function. Then if $P$ and $Q$ are any partitions of $[a,b]$, we have

      1. If $P\subset Q$, then $L(P,f,\alpha )\le L(Q,f,\alpha )$ and $U(Q,f,\alpha )\le U(P,f,\alpha )$

      2. $L(P,f,\alpha )\le U(Q,f,\alpha )$

      3. $\underset{-a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd\alpha \le \underset{a}{\overset{-b}{\mathrm{\int <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd\alpha$.

   2. Let $f:[a,b]\to ℝ$ be increasing. Then $f$ is Riemann integrable on $[a,b]$ iff for each $𝜖>0$, there is a partition $P$ such that $U(P,f,\alpha )-L(P,f,\alpha )\le 𝜖$.

   3. If $f:[a,b]\to ℝ$ is monotone and $\alpha :[a,b]\to ℝ$ is increasing and continuous, then $f$ is Riemann-Stieltjes integrable on $[a,b]$.

   4. If $f:[a,b]\to ℝ$ is continuous and $\alpha :[a,b]\to ℝ$ increasing, then $f$ is Riemann-Stieltijes integrable on $[a,b]$.

   5. Fundamental Theorem of Integral Calculus: If $f:[a,b]\to ℝ$ is differentiable on $[a,b]$ and $f^{\prime }$ is Riemann integrable on $[a,b]$ then $\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}{f}^{\prime }dx=f\left(b\right)-f\left(a\right)$.

   6. If $f_1,f_2:[a,b]\to ℝ$ are bounded, $\alpha :[a,b]\to ℝ$ is increasing and $f_1,f_2$ are Riemann integrable with respect to $\alpha$ on $[a,b]$, then

      1. For any real numbers $c_1,c_2,c_1{f}_1+c_2{f}_2$ is also Riemann integrable on $[a,b]$ and $\begin{array}{ccc}\hfill \underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}(c_1{f}_1& +c_2{f}_2)d\alpha \hfill & \hfill =c_1\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}{f}_{1}d\alpha +c_2\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}{f}_{2}d\alpha .\end{array}$

      2. If $f_1\left(x\right)\le f_2\left(x\right)$ for all $x\in [a,b]$, then $\begin{array}{c}\hfill \underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}{f}_1\left(x\right)d\alpha \le \underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}{f}_2\left(x\right)d\alpha .\end{array}$

      3. If $m\le f_1\left(x\right)\le M$ for all $x\in [a,b]$, then
         $\begin{array}{ccc}\hfill m\left\{\alpha \right(b)-\alpha (a\left)\right\}& \le \underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}{f}_{1}d\alpha \hfill & \hfill \le M\left\{\alpha \right(b)-\alpha (a\left)\right\}\end{array}$

      4. $\beta :[a,b]\to ℝ$ is increasing and $f$ is Riemann integrable with respect to $\beta$ on $[a,b]$ and $c_1$ and $c_2$ are any non-negative real numbers, then $f$ is Riemann integrable with respect to $(c_1\alpha +c_2\beta )$ on $[a,b]$ and $\begin{array}{ccc}\hfill \underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd(c_1\alpha +c_2\beta )& =c_1\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd\alpha \hfill & \hfill +c_2\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd\beta .\end{array}$

   7. Associative law of integrals: Suppose $f:[a,b]\to ℝ$ is bounded and $\alpha :[a,b]\to ℝ$ is increasing. If $a<c<b$, then $f$ is Riemann- Stieltjes integrable on $[a,b]$ iff $f$ is Riemann-Stieltjes integral on $[a,c]$ and $[c,b]$ and $\begin{array}{c}\hfill \underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd\alpha =\underset{a}{\overset{c}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd\alpha +\underset{c}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd\alpha .\end{array}$

   8. Suppose $f:[a,b]\to [c,d],\alpha :[a,b]\to ℝ$ is increasing. $f$ is Riemann-Stieltjes integrable on $[a,b]$ and $\varphi :[c,d]\to ℝ$ is continuous. Then $\varphi$ is Riemann-Stieltjes integrable on $[a,b]$.

   9. If $f,g:[a,b]\to ℝ,\alpha :[a,b]\to ℝ$ is increasing, $f,g$ is Riemann-Stieltjes integrable on $[a,b]$ then

      1. $fg$ is Riemann-Stieltjes integrable on $[a,b]$.

      2. $\left|f\right|$ is Riemann-Stieltjes integrable on $[a,b]$.

      3. $|\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}fd\alpha |\le \underset{a}{\overset{b}{\mathrm{\int <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|f\right|d\alpha$

   10. First Mean Value Theorem: If $f:[a,b]\to ℝ$ is continuous and $\alpha :[a,b]\to ℝ$ is increasing, then there is $c\in [a,b]$ such that $\begin{array}{c}\hfill {\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}_a^{b}fd\alpha =f\left(c\right)\left[\alpha \right(b)-\alpha (a\left)\right]\end{array}$

   11. If a partition $P$ is finer that partition $Q$, (i.e., $Q\subseteq P$), then
       $\begin{array}{ccc}\hfill L(f,Q)& \le L(f,P)\text{and}U(f,Q)\hfill & \hfill \le U(f,P)\end{array}$

   12. For every pair of partitions $P$ and $Q$, $L(f,P)\le U(f,Q)$

   13. For any partition $P,$ $\begin{array}{ccc}\hfill L(f,P)\le & \underset{-}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f\left(x\right)dx\le \hfill & \hfill \stackrel{-}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f\left(x\right)dx\le U(f,P)\end{array}$

   14. Riemann’s criterion: A bounded function $f:[a,b]\to ℝ$ is Riemann integrable iff for every $𝜖>0$ there exists a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<𝜖$ holds .

   15. Darboux theorem: Let $f:[a,b]\to ℝ$ be Riemann integrable and let $\left\{P_n\right\}$ be a sequence of partitions of $[a,b]$ such that $\lim <!--FUNCTION\; APPLICATION-->\left|P_a\right|=0$. Then $\begin{array}{c}\hfill \underset{n\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}L(f,P_n)=\underset{n\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}U(f,P_n)=\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx\end{array}$

   16. Lebesgue-Vitali theorem: A bounded function $f:[a,b]\to ℝ$ is Riemann integrable iff it is continuous almost everywhere.

   17. The collection of all Riemann integrable functions on a closed interval is a function space and an algebra of functions.

   18. Fundamental theorem of calculus: For a continuous function $f:[a,b]\to ℝ$

       1. If $A:[a,b]\to ℝ$ is an area function of $f$ (i.e., $A\left(x\right)=\underset{c}{\overset{x}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(t\right)dt$ holds for all $x\in [a,b])$. Then $A$ is an anti-derivative of $f$. i.e., $A^{\prime }\left(x\right)=f\left(x\right)$ holds for each $x\in [a,b]$

       2. If $F:[a,b]\to ℝ$ is an anti-derivative of $f$, i.e., $F^{\prime }\left(x\right)=f\left(x\right)$ holds for each $x\in [a,b]$, then $\begin{array}{c}\hfill \underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=F\left(b\right)-F\left(a\right)\end{array}$

1.19 Improper Riemann Integral

1. If $f:[a,\infty ]\to ℝ$ is Riemann integrable on every closed sub-interval of $[a,\infty ]$, then its improper Riemann integral is $\begin{array}{c}\hfill \underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=\underset{s\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}\underset{a}{\overset{s}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx\end{array}$

2. Some Important Theorems

   1. Assume $f:[a,\infty ]\to ℝ$ is Riemann integrable on every closed subinterval of $[a,\infty ]$. Then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ exists iff every $𝜖>0$ there exists some $M>0$ (depending on $𝜖$) such that $|\underset{s}{\overset{t}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx|<𝜖$ for all $s,t\ge M.$

   2. If a function $f:[a,b]\to ℝ$ is Riemann integrable on every closed subinterval of $[a,\infty ]$ and $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|f\right(x\left)\right|dx$ then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ also exists and $|\underset{s}{\overset{t}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx|\le \underset{s}{\overset{t}{\mathrm{\int <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|f\right(x\left)\right|dx.$

   3. Let $f:[a,\infty ]\to ℝ$ be Riemann integrable on every closed subinterval of $[a,\infty ]$. Then $f$ is Lebesgue integrable iff the improper Riemann integral $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|f\right(x\left)\right|dx$ exists. Moreover $\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->fd\lambda =\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$.

   4. Euler theorem: [1Em] $\underset{0}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}{e}^{-x^2}d\lambda =\frac{\sqrt{\pi }}{2}$

   5. For each $t\in ℝ$,[1Em] $\underset{0}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}{e}^{-x^2}\cos <!--FUNCTION\; APPLICATION-->\left(2xt\right)dx=\frac{\sqrt{\pi }}{2}{e}^{-t^2}$

   6. If $t\ge 0$, then [1Em] $\underset{0}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\frac{\sin <!--FUNCTION\; APPLICATION-->x}{x}{e}^{-xt}dt=\frac{\pi }{2}-\arctan <!--FUNCTION\; APPLICATION-->t$

1.20 Types of Improper Integrals

1. $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$

2. $\underset{-\infty }{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$

3. $\underset{a+}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ and $\underset{n\to a+}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)$ does not exits.

4. $\underset{a}{\overset{b-}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ and $\underset{n\to b-}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)$ does not exits.

Type I. $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$

1. Convergence of integral: The integral $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ converges iff $\underset{s\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}\underset{a}{\overset{s}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=A$ exist and $A$ is the value of integral.

2. Divergence of integral: The integral $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ is divergent, if it is not convergent.

3. Absolute convergence: The integral $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ converges absolutely then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|f\right(x\left)\right|dx$ converges.

4. Conditionally convergence: The integral $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ converges conditionally if integral but not absolute converges.

5. Sum of integral:

   1. The sum of number of finite improper integral converges iff each of these integrals converges.

   2. The sum of number of finite improper integral diverges iff one of these integral diverges.

6. Some Important Theorems

   1. Comparison test:

      1. If $\begin{array}{cccc}\hfill f\left(x\right)& \le g\left(x\right)\text{then}\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}g\left(x\right)dx\hfill & \hfill <\infty \Rightarrow \underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx& <\infty \hfill \end{array}$

      2. If $\begin{array}{cccc}\hfill f\left(x\right)& \le g\left(x\right)\text{then}\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}g\left(x\right)dx\hfill & \hfill =\infty \Rightarrow \underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx& =\infty \hfill \end{array}$

   2. If $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|f\right(x\left)\right|dx<\infty$ then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ converges.

   3. If $\underset{x\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}\frac{f\left(x\right)}{g\left(x\right)}$ exist and $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|g\right(x\left)\right|dx<\infty$ then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ converges to absolutely.

   4. Limit test for convergence: If $\underset{x\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}{x}^{p}f\left(x\right)=A.(p>1)$ then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|f\right(x\left)\right|dx<\infty$.

   5. If $\underset{x\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}x{(\log <!--FUNCTION\; APPLICATION-->x)}^{p}f\left(x\right)=A.(p>1)$ then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|f\right(x\left)\right|dx<\infty$.

   6. Limit test for divergence: If $\underset{x\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}xf\left(x\right)=A\ne 0$ (or $\pm \infty .)$ then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ diverges. The test fails at $A=0$.

   7. If $\underset{x\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}x(\log <!--FUNCTION\; APPLICATION-->x)f\left(x\right)=A\ne 0$ (or $\pm \infty .)$ then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ diverges.

   8. Ig $g\left(x\right)$ is non decreasing function $\underset{x\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}g\left(x\right)=0$

      1. $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}g\left(x\right)\sin <!--FUNCTION\; APPLICATION-->xdx$ converges.

      2. $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}g\left(x\right)dx=\infty$ then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}g\left(x\right)|\sin <!--FUNCTION\; APPLICATION-->x|dx=\infty .$

      3. And $n$ is an integer $>\frac{a}{\pi }$ then $|\underset{n\pi }{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}g\left(x\right)\sin <!--FUNCTION\; APPLICATION-->xdx|\le 2g\left(n\pi \right).$

Type II. $\underset{-\infty }{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$

The integral $\underset{-\infty }{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ becomes $\underset{-b}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f(-t)dt$ when

1. If $\underset{t\to +\infty }{\lim <!--FUNCTION\; APPLICATION-->}f(-t)t^p=\underset{x\to -\infty }{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right){(-x)}^p=A$ also if $(p>1)$ then $\underset{-\infty }{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ converges absolutely.

2. If $\underset{t\to +\infty }{\lim <!--FUNCTION\; APPLICATION-->}f(-t)t=\underset{x\to -\infty }{\lim <!--FUNCTION\; APPLICATION-->}-f\left(x\right)x=A\ne 0$ (or $\pm \infty$ ) then the integral $\underset{-\infty }{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ diverges.

Type III. $\underset{a^{+}}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ and $\underset{x\to a^{+}}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)$ does not exist.

1. Convergence of integral: The integral $\underset{a^{+}}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ converges iff $\underset{𝜖\to 0^{+}}{\lim <!--FUNCTION\; APPLICATION-->}\underset{a+𝜖}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=A$ exist and $A$ is the value of integral.

2. Divergence of integral: The integral $\underset{a^{+}}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ is diverges iff it does not converge.

3. Absolute convergence: The integral $\underset{a^{+}}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ converges absolutely if $\underset{a^{+}}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|f\right(x\left)\right|dx$ converges.

4. Conditionally convergence: The integral $\underset{a^{+}}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ converges but the integral $\underset{a^{+}}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\left|f\right(x\left)\right|dx$ diverges.

Type IV. $\underset{a}{\overset{b^{-}}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ and $\underset{x\to b^{-}}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)$ does not exist.

The integral $\underset{a}{\overset{b^{-}}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ becomes $\underset{a^{+}}{\overset{b-a}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f(b-t)dt$ when we set $x=b-t$

1. The integral $\underset{a}{\overset{b^{-}}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ converges absolutely if $\underset{t\to 0^{+}}{\lim <!--FUNCTION\; APPLICATION-->}f(b-t)t^p=\underset{x\to b^{-}}{\lim <!--FUNCTION\; APPLICATION-->}{(b-x)}^{p}f\left(x\right)=A$ $(0<p<1)$.

2. The integral $\underset{a}{\overset{b^{-}}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$ diverges if $\underset{t\to 0^{+}}{\lim <!--FUNCTION\; APPLICATION-->}f(b-t)t=\underset{x\to b^{-}}{\lim <!--FUNCTION\; APPLICATION-->}(b-x)f\left(x\right)=A\ne 0$.

1.21 Uniform Convergence

1. The integral $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f(x,t)dt$ converges uniformly to $F\left(x\right)$ in the interval $x\in [A,B]$ iff for arbitrary $𝜖>0$ corresponds a number $Q$ independent of $x\in [A,B]$ such that when $R>Q.$ Then

   $$|F\left(x\right)-{\int \; <!--nolimits-->}_a^{R}f(x,t)dt|<𝜖.$$

2. The integral $\underset{a^{+}}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f(x,t)dt$ uniformly to $F\left(x\right)$ in the interval $x\in [A,B]$ iff for arbitrary $𝜖>0$ corresponds a number $Q$ independent of $x\in [A,B]$ such that when $a<R<b$, then

   $$|F\left(x\right)-{\int \; <!--nolimits-->}_R^{b}f(x,t)dt|<𝜖.$$

Important Theorem $\begin{array}{ccc}\hfill \text{If}\left|f\right(x,t\left)\right|\le M\left(T\right),t& \in (a,\infty ),\text{and}x\hfill & \hfill \in [A,B]\end{array}$

1. Then $\underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}M\left(t\right)dt=\infty$ $\Rightarrow \underset{a}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f(x,t)dt$ converges uniformly in $x\in [A,B]$.

2. Then $\underset{a^{+}}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}M\left(t\right)dt<\infty$ $\Rightarrow \underset{a^{+}}{\overset{\infty }{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f(x,t)dt$ converges uniformly in $x\in [A,B]$.

1.22 Discontinuities of Real Valued Function

1. Continuity: A function $f\left(x\right)$ is called continuous at $x=a$, if

   1. $f\left(x\right)$ is defined at $x=a$, i.e., $f\left(x\right)=f\left(a\right)$ at $x=a.$

   2. $\underset{x\to a^{+}}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)$ and $\underset{x\to a^{-}}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)$ exist.

   3. $f\left(a^{+}\right)=f\left(a^{-}\right)=f\left(a\right)$.

2. Discontinuity: A function $f\left(x\right)$ which is not continuous at $x=a$, is called discontinuous at $x=a$.

3. Types of Discontinuities

   1. Discontinuity of first kind: A function $f\left(x\right)$ has discontinuity of first kind at $x=a$, if left and right hand limit exists at $x=a$, but they are distinct. $f\left(a^{+}\right)\ne f\left(a^{-}\right)$.

   2. Discontinuity of second kind: A function $f\left(x\right)$ has discontinuity of second kind at $x=a$, if left and right hand limit does not exist, i.e., neither $f{(}^{+})$ nor $f\left(a^{-}\right)$ exist.

   3. Mixed discontinuity: A function $f\left(x\right)$ has mixed discontinuity at $x=a$ if either of left or right hand limit exist.

   4. Removable discontinuity: A function $f\left(x\right)$ has the removable discontinuity at $x=a$ if $f\left(a^{+}\right)$ and $f\left(a^{-}\right)$ exist but $f\left(a^{+}\right)=f\left(a^{-}\right)\ne f\left(a\right).$

   5. Irremovable discontinuity: A function $f\left(x\right)$ has irremovable discontinuity at $x=a$ if $f$ has discontinuity of first kind, second kind or mixed discontinuity.

   6. Jumps and jump discontinuity: If $f\left(a^{+}\right)$ and $f\left(a^{-}\right)$ exist at $x=a$, then

      1. $f\left(a\right)-f\left(a^{-}\right)$ is called left hand jump of $f$ at $a$.

      2. $f\left(a^{+}\right)-f\left(a\right)$ is called right hand jump of $f$ at $a$.

      3. $f\left(a^{+}\right)-f\left(a^{-}\right)$ is called jump of $f$ at $a$.

      1. If any of these jumps is different from 0, then $a$ is called the jump discontinuity of $f.$

      2. Jump discontinuity are discontinuity of first kind.

   7. Infinite discontinuity: A function $f\left(x\right)$ has infinite discontinuity at $x=a$, if any of the four functional limits $f\left(a^{+}\right),f\left(a^{-}\right),\overline{f\left(a^{+}\right)},\underline{f\left(a^{-}\right)}$ are indefinitely large or infinite.

   8. Saltus (Measure of discontinuity): The saltus of a function $f\left(x\right)$ at $x=a$ is the greatest positive difference between any two of the five numbers $\overline{f\left(a^{+}\right)},\underline{f\left(a^{+}\right)},\overline{f\left(a^{-}\right)},\underline{f\left(a^{-}\right)}$ and $f\left(a\right).$

      1. Saltus on rightSaltus on right: The greatest positive difference between any two of the three, $\overline{f\left(a^{+}\right)},\underline{f\left(a^{+}\right)}$ and $f\left(a\right).$

      2. Saltus on rightSaltus on left: The greatest positive difference between any two of the three, $\overline{f\left(a^{-}\right)},\underline{f\left(a^{-}\right)}$ and $f\left(a\right).$

      3. Important: Saltus is zero at point of continuity and greater then zero at point of discontinuity.

1.23 Monotonic Functions

$f:E\to ℝ,E\subset ℝ$

1. Increasing function: A function $f$ is increasing function (non-decreasing) on $E$ if $\forall <!--FUNCTION\; APPLICATION-->x,y\in E,x<y\Rightarrow f\left(x\right)\le f\left(y\right).$

2. Strictly increasing function: A function $f$ is strictly increasing function on $E$ if $\forall <!--FUNCTION\; APPLICATION-->x,y\in E,x<y\Rightarrow f\left(x\right)<f\left(y\right).$

3. Decreasing function: A function $f$ is decreasing function (non-increasing) on $E$ if $\forall <!--FUNCTION\; APPLICATION-->x,y\in E,x<y\Rightarrow f\left(x\right)\ge f\left(y\right).$

4. Strictly decreasing function: A function $f$ is strictly decreasing function on $E$ if $\forall <!--FUNCTION\; APPLICATION-->x,y\in E,x<y\Rightarrow f\left(x\right)>f\left(y\right).$

5. Monotonic function: A function that is either monotonic increasing or decreasing function is called monotonic function.

6. Singular monotonic function: A monotonic function $f$ on $[a,b]$ such that $f^{\prime }\left(x\right)=0,\forall <!--FUNCTION\; APPLICATION-->x\in [a,b].$

Some Important Theorems

1. If $f$ is increasing function then $-f$ is decreasing function.

2. If $f$ is increasing function on closed interval $[a,b]$, then $f\left(c^{+}\right)$ and $f\left(c^{-}\right)$ exist for each $c\in (a,b)$ and $f\left(c^{-}\right)\le f\left(c\right)\le \left(f\right(c^{+})$.

3. If $f$ is increasing on $[a,b]$ then at end points. $f\left(a\right)\le f\left(a^{+}\right)$ and $f\left(b^{-}\right)\le f\left(b\right)$.

4. If $f$ is strictly increasing on $E\subset ℝ$. Then $f^{-1}$ exist and is strictly increasing on $f\left(E\right)$.

5. If $f$ is one-to-one and continuous on $[a,b]$, then $f$ is strictly monotonic on $[a,b]$.

6. Of non-decreasing function (increasing function) every discontinuous point is of first kind.

7. If $f$ is increasing function on $[a,b]$ and $x_0,x_1,\dots <!--FUNCTION\; APPLICATION-->,x_n$ are $n+1$ points such that $a=x_0<x_1<\dots <!--FUNCTION\; APPLICATION--><x_n=b$. Then we have $\underset{k=1}{\overset{n-1}{\sum <!--FUNCTION\; APPLICATION-->}}\left[f\right(x_{k^{+}})-f(x_{k^{-}}\left)\right]\le f\left(b\right)-f\left(a\right)$.

8. If $f$ is continuous on $[a,b]$ then the set of discontinuous of $f$ is countable.

9. If $f$ is continuous on $[a,b]$ and $f$ exist (finite or infinite) at each point of the interval $(a,b)$. Then

   1. If $f>0$ in $(a,b)$. $f$ is strictly increasing on $[a,b]$.

   2. If $f<0$ in $(a,b)$. $f$ is strictly decreasing on $[a,b]$.

   3. $f^{\prime }=0$ everywhere in $(a,b)$, then $f$ is constant on $[a,b]$.

10. If $f$ has derivative (finite or infinite) on $(a,b)$ and $f$ is continuous on $[a,b]$. If $f^{\prime }\left(x\right)\ne 0$, then $\forall <!--FUNCTION\; APPLICATION-->x\in [a,b]$, $f$ is strictly monotonic on $[a,b]$.

11. If $f$ exist and is monotonic on $(a,b)$ then $f^{\prime }$ is continuous on $[a,b]$.

1.24 Functions of Bounded Variation

1. Partition: If $[a,b]$ is a compact interval, a set of points $P=\{x_0,x_1,\dots <!--FUNCTION\; APPLICATION-->,x_n\}$ satisfying the inequalities,$a=x_0<,\dots <!--FUNCTION\; APPLICATION-->,x_n=b$, is called a partition of $[a,b]$. For k-th subinterval, $[x_{k-1},x_k]$ and $\mathrm{\Delta }{x}_k=x_k-x_{k-1}$ for $P$ then we have $\underset{k-1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\mathrm{\Delta }{x}_k=b-a$.

2. Bounded variation: Let $f$ be defined on $[a,b]$ and $P=\{x_0,x_1,\dots <!--FUNCTION\; APPLICATION-->,x_n\}$ be a partition of $[a,b]$. If $\mathrm{\Delta }{f}_k=f\left(x_k\right)-f\left(x_{k-1}\right),k=1,2,3,\dots <!--FUNCTION\; APPLICATION-->,n.$ Then for positive integer $M$ such that $\underset{k-1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\left|\mathrm{\Delta }{f}_k\right|\le M$ for all partitions of $[a,b]$, $f$ is of bounded variation on $[a,b]$.

3. Total Variation: If $f$ is of bounded variation on $[a,b]$ and if $\sum <!--FUNCTION\; APPLICATION-->\left(P\right)$ denotes the sum, $\sum <!--FUNCTION\; APPLICATION-->P=\underset{k=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\left|\mathrm{\Delta }{f}_k\right|$, corresponding to partition $P=\{x_0,x_1,\dots <!--FUNCTION\; APPLICATION-->,x_n\}$ of $[a,b]$. Then the number, $V_f(a,b)=\sup <!--FUNCTION\; APPLICATION-->\{\sum <!--FUNCTION\; APPLICATION-->(P):P\text{is a partition}\}$ is called the total variation of $f$ on the interval $[a,b].$

   1. $V_f(a,b)$ is finite number, since $f$ is of bounded variation on $[a,b]$.

   2. $V_f(a,b)\ge 0$ since $\sum <!--FUNCTION\; APPLICATION-->\left(P\right)\ge 0$.

   3. $V_f(a,b)=0$ iff $t$ is constant on $[a,b]$.

Some Important Theorems

1. If $f$ is monotone on $[a,b]$, then $f$ is of bounded variation on $[a,b]$.

2. If $f$ is continuous on $[a,b]$, $f^{\prime }$ exists and $\left|f^{\prime }\right(x\left)\right|\le M,\forall <!--FUNCTION\; APPLICATION-->x\in (a,b)$ then $f$ is of bounded variation on $[a,b]$.

3. If $f$ is of bounded variation on $[a,b]$. $\left|f^{\prime }\right(x\left)\right|\le M,\forall <!--FUNCTION\; APPLICATION-->x\in (a,b)$ for all partition of $[a,b]$, then $f$ is bounded on $[a,b]$ and $\left|f\right(x\left)\right|\le \left|f\right(a\left)\right|+M,x\in [a,b]$.

4. If $f$ and $g$ are of bounded variation on $[a,b]$. Then there sum, difference and product are also of bounded on $[a,b]$ i.e., $f\pm g$ and $fg$ are bounded variation on $[a,b]$.

5. Quotient of $f∕g$ and $1∕f$ are not of bounded variation, however if $f,g$ are of bounded variation.

6. If $f$ is of bounded variation on $[a,b]$ and $f$ is bounded above from zero. i.e., for $m>0$, $0<m\le \left|f\right(x\left)\right|$ for all $x\in [a,b]$. Then $1∕f$ is also of bounded variation on $[a,b]$. If $g=\frac{1}{f}$ then $V_g(a,b)\le \frac{V_f(a,b)}{m^2}$.

7. If $f$ is of bounded variation on $[a,b]$ and $c\in (a,b)$. Then $f$ is of bounded variation on $[a,c]$ and on $[c,b]$ and also $V_f(a,b)=V_f(a,c)+V_f(c,b)$.

8. If $f$ is of bounded variation on $[a,b]$. Let $V$ be defined on $[a,b]$ as follows $V\left(x\right)=V_f(a,x)$, $x\in (a,x],V\left(a\right)=0$. Then

   1. $V$ is an increasing function on $[a,b]$.

   2. $V-f$ is an increasing function on $[a,b]$.

9. A function on $f$ is of bounded variation on $[a,b]$ iff $f$ is the difference of two monotone real valued functions on $[a,b]$.

10. If $f$ is of bounded variation on $[a,b]$ and $V\left(x\right)=V_f(a,x),\forall <!--FUNCTION\; APPLICATION-->x\in (a,b]$, $V\left(a\right)=0$. Then

    1. Every point of continuous of $f$ is also a point of continuity of $V$.

    2. Every point of continuity of $V$ is also a point of continuity of $f$.

11. If $f$ is continuous on $[a,b]$. Then $f$ is of bounded variation on $[a,b]$ iff $f$ is the difference of two monotone continuous functions on $[a,b]$.

12. If $f$ is integrable on $[a,b]$, then function $F\left(x\right)=\underset{a}{\overset{x}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(t\right)dt$ is continuous function of bounded variation on $[a,b]$.

1.25 Absolutely Continuous Function

1. Absolutely continuous: A real-valued function $f$ defined on $[a,b]$ is absolutely continuous on $[a,b]$ if for every $𝜖>0,\exists <!--FUNCTION\; APPLICATION-->\delta >0$ such that $\underset{k=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\left|f\right(b_k)-f(a_k\left)\right|<𝜖$ for every $n$ disjoint open sub-intervals $(a_k,b_k)$ of $[a,b],n=1,2,\dots <!--FUNCTION\; APPLICATION-->$, $\underset{k=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}(b_k-a_k)<\delta$.

Some Important Theorems

1. Every absolutely continuous function on $[a,b]$ is continuous and of bounded variation on $[a,b]$.

2. If $f$ and $g$ are absolutely continuous on $[a,b]$ then $\left|f\right|,cf$ ($c$ is a constant), $f+g,fg$ are absolutely continuous on $[a,b],f∕g$ is absolutely continuous on $[a,b]$ if $g$ is bounded away from zero.

3. If $f$ is absolutely continuous on $[a,b]$ and $f\left(x\right)=0$ then $f$ is constant almost everywhere.

4. If $f$ is absolutely continuous, then $f$ has a derivative almost everywhere.

5. A function $F$ is indefinite integral iff it is absolutely continuous.

6. Every absolutely continuous function is the indefinite integral of its derivative.

1.26 Elements of Metric Spaces

1. Metric spaces: Let $X$ be a non-empty set and we have a real valued function $d:X\times x\to ℝ$ such that for all $x,y\in X$,

   1. $d(x,y)\ge 0$.

   2. $d(x,y)=0$ iff $x=y$.

   3. $d(x,y)\le d(x,z)+d(z,y)$ Then $d$ is called metric and $(X,d)$ is a metric space.

2. Pseudo metric: A metric $d$ with the relaxation, $d(x,y)=0$ for some $x\ne y$.

3. Extended metric: A metric $d$ such that $d:X\times X\to ℝ\cup \left\{\infty \right\}$.

4. Discrete metric: $X$ is any set, $\begin{array}{c}\hfill d(x,y)=\{\begin{array}{cc}0\hfill & \text{if}x=y\hfill \\ 1\hfill & \text{if}x\ne y\hfill \end{array}\end{array}$

5. Subspace: $(X,d)$ be a metric space, $Y\subset X$ then $(Y,d)$ is a subspace of $(X,d)$.

6. Euclidean metric: $\overline{x},\overline{y}\in X\subset {ℝ}^n$ $\overline{x}=(x_1,x_2,\dots <!--FUNCTION\; APPLICATION-->,x_n),\overline{y}=(y_1,y_2,\dots <!--FUNCTION\; APPLICATION-->,y_n)$ $d(\overline{x},\overline{y})=[\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}|x_i-y_i|{]}^{\frac{1}{n}}$.

7. Convergence: A sequence $\left\{x_n\right\}$ converges to an element $x\in X$ if $\underset{n\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}d(x_n,x)=0$.

8. Open sphere (Open ball): $S(x_0,r)=\{x\in X:d(x_0,x)<r\},r>0,x_0\in X.$

9. Closed sphere (Closed ball): $\overline{S}(x_0,r)=\{x\in X:d(x_0,x)\le r\},r>0,x_0\in X.$

10. Open set: A set $G\subset X$ is open if it contains a sphere about each of its points or every point is an interior point.

11. Closed set: A set $F$ is closed if $X-F$ is open or if every point is a limit point.

12. Neighbourhood: A neighbourhood of a point $x\in X$ is an open set which contains $x_0.$

13. Interior point: $x_0$ is an interior point of set $A$ if $A$ is a neighbourhood of $x_0$.

14. Interior set: Interior of set $A$ contains all interior points of $A$.

15. Limit point: If $A\subset X$ and $x_0\in X$, then $x_0$ is limit point of $A$, if every neighbourhood of $x_0$ contains point of $A$ distinct from $x_0$.

16. Closure: Closure of a set $A$ is $\overline{A}$ which contains all points which are either point of $A$ or limit point of $A$.

17. Dense set: $A,B\subset X$. $A$ is dense in $B$ if $B\subset \overline{A}$.

18. Everywhere dense: $A$ is everywhere dense if $\overline{A}=X.$

19. Separable metric space: If metric space $X$ has a countable subset which is everywhere dense, then it is separable metric space.

20. Open covering: If $𝒞$ is a collection of open sets in metric space $X$ with the property that every $x\in X$ is a member of atleast one set $G\in 𝒞$. The $𝒞$ is called the open covering of $X$.

21. Sub-covering: A sub-covering of the open covering $𝒞$ is any collection ${𝒞}^{\prime }\subset 𝒞$ which is also open covering of $X$.

22. Cauchy sequence: A sequence $\left\{x_n\right\}$ in a metric space $X=(X,d)$ is Cauchy sequence, if for every $𝜖>0$, there is $N$ such that $m,n>N\Rightarrow d(x_m,x_n)<𝜖.$

23. Complete metric space: A metric space is complete, if every Cauchy sequence in metric space converges.

24. Contraction: If $(X,d)$ is a metric space, a mapping $T:X\to X$ is called contraction in $X$ if there is a constant $K$, with $0\le K<1$, such that $x,y\in X,x\ne y$ $\Rightarrow d(Tx,Ty)\le Kd(x,y).$

25. Lipschitz condition: For Lipschitz constant $M>0$, $\left|f\right(x,y_2)-f(x,y_1\left)\right|\le M|y_2-y_1|$.

26. Completion: The enlarged space is called completion of metric space $X.$

27. Isometry: $T:(X,d)\to (Y,\sigma )$, such that $\forall <!--FUNCTION\; APPLICATION-->x,y\in X,$ $\sigma (Tx,Ty)=d(x,y).$

28. Bolzano-Weierstrass property: A space $X$ has Bolzano-Weierstrass property if every infinite sequence in $X$ has atleast one limit point.

29. Nowhere dense: If $A\subset X$ is nowhere dense, $\overline{A}$ the closure of $A$ has no interior point.

30. First category: A set $A\subset X$ is of first category in $X$, if it is the union of countably many nowhere dense sets in $X$.

31. Second category: A set $A\subset X$ is of second category in $X$, if it is not of first category.

32. Compact metric space:

    1. A metric space $(X,d)$ is compact if every infinite subset of $X$ has atleast one limit point.

    2. A metric space $X$ is compact if every open covering $\mathscr{H}$ of $X$ has a finite sub-covering. Set $K\subset X$ is compact if $(K,d)$ is compact.

33. Relatively compact: If $X$ is a metric space. $K\subset X$ and closure of $K,\overline{K}$ is compact, then $K$ is relatively compact to $X.$

34. Total boundedness: A metric space $X$ is totally bounded if for every $𝜖>0,X$ contains a finite set, called an $𝜖$-net, such that the finite set of open spheres of radius $𝜖$ and centres in the $𝜖$-net covers $X$.

35. Continuity: If $T:(X,d)\to (Y,d).T$ is continuous at $x\in X$ if every sequence $\left\{x_n\right\}$ converges to $x,\left\{T{x}_n\right\}$ converges to $Tx\in Y.$ $T$ is continuous if it is continuous a every $x\in X.$

36. Uniformly continuity: If $T:(X,d)\to (Y,d).T$ is uniformly continuous on $X$,if for every $𝜖>0$, there is $\delta >0$ such that $d(x,x^{\prime })<\delta \Rightarrow d(Tx,T{x}^{\prime })<𝜖,$ for all $x,x^{\prime }\in X.$

37. Connected set: $A\subset X$, set $A$ is called connected set if it cannot be represented as the union of two sets, each of which is disjoint from the closure of the other.

38. Uniformly boundedness: A collection $ℱ$ of function on set $X$ is uniformly bounded if there is $M>0$ such that $\left|f\right(x\left)\right|\le M$ for all $x\in X$ and all $f\in ℱ.$

39. Equicontinuous: A collection $ℱ$ of functions defined on a metric space $X$ is equicontinuous if for each $𝜖>0$ there is $\delta >0$ such that $d(x,x^{\prime })<\delta$ $\Rightarrow \left|f\right(x)-f(x^{\prime }\left)\right|<𝜖$ for all $x,x^{\prime }\in X$ and $f\in ℱ$.

40. Algebra: $(X,d)$ is compact metric space, $C\left(X\right)$ is the space of continuous real functions on $X$, such that $d(f,g)=\max <!--FUNCTION\; APPLICATION-->\left\{\right|f\left(x\right)-g\left(x\right)|:x\in X\}$. Set $A\subset C\left(X\right)$ is called an algebra if $f,g\in A$ and $a$ any number $f+g\in A,fg\in A$ and $af\in A$

41. Algebra generated by set $E$: If $E\subset C\left(X\right)$ the intersection of all algebras in $C\left(X\right)$ containing $E$, which is itself an algebra containing $E$ is called algebra by $E.$

Some Important Results and Theorems

1. Holder inequality: If $p>1$ and $\frac{1}{p}+\frac{1}{q}=1$ and $\overline{x}=(x_1,x_2,\dots <!--FUNCTION\; APPLICATION-->,x_n),\overline{y}=(y_1,y_2,\dots <!--FUNCTION\; APPLICATION-->,y_n)\in {ℝ}^n$ $\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\left|x_i{y}_i\right|\le [\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\left|x_i\right|{]}^{\frac{1}{p}}[\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\left|y_i\right|{]}^{\frac{1}{q}}$

2. Minkowski inequality: If $p\ge 1$, and $\overline{x},\overline{y}\in {ℝ}^n$ $[\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}|x_i+y_i|{]}^{\frac{1}{p}}\le [\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\left|x_i{|}^p{]}^{\frac{1}{p}}[\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\right|y_i{|}^p{]}^{\frac{1}{q}}$

3. Cauchy- Schwarz inequality: Substituting $p=q=\frac{1}{2}$, we have $[\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}|x_i+y_i|{]}^2\le [\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\left|x_i{|}^2][\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}\right|y_i{|}^2]$

4. The collection $𝒞$ of all open set satisfies:

   1. $\varnothing ,X\in 𝒞$.

   2. Any union of members of $mathcalC$ is a member of $𝒞$.

   3. The intersection of finitely many members of $𝒞$ is a member of $𝒞$.

5. The collection $ℱ$ of closed sets satisfies:

   1. $\varnothing ,X\in ℱ$.

   2. Any intersection of closed set is closed.

   3. A finite union of closed sets is closed.

6. The interior of a set $A$ is the largest open set contained in $A$.

7. The closure of set $A$ is the smallest closed set containing $A.$

8. Set $A\subset X$ is everywhere dense if $\forall <!--FUNCTION\; APPLICATION-->x\in X,𝜖>0$, there is $y\in A:d(x,y)<𝜖.$

9. $X$ is separable iff there is a countable collection $B$ of open sets such that an arbitrary open set can be expressed as a union of members of $B$.

10. Lindelof: If $X$ is separable and $𝒞$ is an open covering of $X$, then there is a countable sub-covering ${𝒞}^{\prime }\subset 𝒞$.

11. A metric $(X,d)$ is complete iff for every sequence $\left\{{\overline{S}}_n\right\}$ of closed spheres, with ${\overline{S}}_{n+1}\subset {\overline{S}}_n$ and $\lim <!--FUNCTION\; APPLICATION-->r_n=0$, where $r_n$ is the radius of ${\overline{S}}_n.$ Then the intersection $\underset{n=1}{\overset{\infty }{\bigcap <!--FUNCTION\; APPLICATION-->}}{\overline{S}}_n$ consists of exactly one point.

12. A subspace of a complete metric space is complete iff it is closed.

13. If $X$ is a complete metric space and $T$ is a contraction in $X$, then $T$ has a fixed point and it is unique.

14. If $f$ is continuous on an open connected set $D$ and satisfies a Lipschitz condition in $y$ on $D$, then for every $(x_0,y_0)\in D$, the differential equation $\frac{dy}{dx}=f(x,y)$ has a unique local solution passing through $(x_0,y_0)$.

15. If $(X,d)$ is a metric space, it can be imbedded as a dense subspace, in a complete metric space $(\tilde{X},\tilde{d})$.

16. Metric space $(X,d)$ is of second category in itself iff any representation of $X$ as union $X=\underset{i=1}{\overset{\infty }{\bigcup <!--FUNCTION\; APPLICATION-->}}{F}_i$ of countable many closed set $F_i$ atleast one of the $F_i$ contains a sphere.

17. Every complete metric space $(X,d)$ is of second category in itself.

18. $K$ is compact iff every sequence with values in $K$ has a subsequence which converges to a point in $K.$

19. Every compact set in a metric space $X$ is closed, bounded subset of $X$.

20. If metric space $X$ is compact, then every closed subset of $X$ is compact.

21. Every compact metric space is complete.

22. Metric space $X$ is bounded iff every sequence in $X$ has a Cauchy subsequence.

23. Every totally bounded metric space is separable.

24. Every compact metric space is separable.

25. A metric space $X$ is compact iff for every collection $𝒞$ of open sets which covers $X$ there are finite subsets $G_1,G_2,\dots <!--FUNCTION\; APPLICATION-->,G_n\in 𝒞$ which covers $X$.

26. Let $T:(X,d)\to (Y,\sigma )$

    1. If $X$ is compact, then every $T$ which is continuous on $X$ is uniformly continuous.

    2. If $T$ is continuous then the image of a compact set is compact.

    3. $T$ is continuous iff for open ball $G\subset Y$ the inverse image $T^{-1}\left(G\right)=\{x:x\in X,Tx\in G\}$ is open in $X$.

27. Arzela-Ascoli theorem: If $X$ is a compact metric space, a subset $K\subset X$ is relatively compact iff it is uniformly bounded and equicontinuous.

28. If $f$ is continuous on an open set $D$, then for every $(x_0,y_0)\in D$ the differential equation $\frac{dy}{dx}=f(x,y)$ has a local solution passing through $(x_0,y_0).$

29. Stone-Weierstrass theorem: If $A$ is closed algebra in $C\left(X\right)$. $X$ has a compact metric space such that $1\in A$ and if $x,y\in X,x\ne y$, there is an $f\in A$ for which $f\left(x\right)\ne f\left(y\right)$. Then $A=C\left(X\right).$

1.27 Lebesgue Measure

1.27.1 Measure

1. Length of an interval: The length of an interval $I$ is the difference of end points of the interval.

2. Measure of a set: Let $ℳ$ be a collection of sets of real numbers and $E\in ℳ$. Then non-negative extended real number $mE$ is called the measure of $E$. If $m$ satisfies

   1. $mE$ is defined for each set $E$ of real numbers. ie., $ℳ=P\left(R\right)$, the power set of sets of real number.

   2. For an interval $I,mI=l\left(I\right)$.

   3. If $\left\{E_n\right\}$ is a sequence of disjoint sets (for which $m$ is defined) in $ℳ.$ $m(\cup E_n)=\sum <!--FUNCTION\; APPLICATION-->m{E}_n$

   4. $m$ is translation invariant. i.e., If $E$ is the set on which $m$ is defined,and $E+y=\{x+y:x\in E\}$ then $m(E+y)=mE$

3. Countable additive measure: Let $ℳ$ be a $\sigma$-algebra of sets of real numbers and $E\in ℳ$. Then non-negative extended real number $mE$ is countable additive measure. If $m(\cup E_n)=\sum <!--FUNCTION\; APPLICATION-->m{E}_n$ for each sequence $\left\{E_n\right\}$ of disjoint sets in $ℳ$.

4. Countable sub additive measure: Let $ℳ$ be a $\sigma$-algebra of sets of real numbers and $E\in ℳ$. Then non-negative extended real number $mE$ is countable sub additive measure, if $m(\cup E_n)\le \sum <!--FUNCTION\; APPLICATION-->m{E}_n$ for each sequence $\left\{E_n\right\}$ of sets in $ℳ$.

Some Important Results Let $m$ be a countable additive measure defined for all sets in a $\sigma$-algebra, $ℳ$. Then

1. Monotonicity: $A,B\in ℳ,A\subset B\Rightarrow mA\le mB$.

2. If for some set $A\in ℳ,mA<\infty$, then $m\varnothing =0.$

1.27.2 Lebesgue Outer Measure

Lebesgue outer measure: Let $A$ be a set of real numbers. $\left\{I_n\right\}$ be the countable collection of open intervals that covers $A$, i,e., $A\subset \cup I_n$. Then Lebesgue outer measure $m^{\ast }A$ of $A$ is $m^{\ast }A=\underset{A\subset \cup I_n}{\inf <!--FUNCTION\; APPLICATION-->}l\left(I_n\right)$. Some Important Results

1. $m^{\ast }\varnothing =0$.

2. $A\subset B\Rightarrow m^{\ast }A\le m^{\ast }B$.

3. For singleton set $\left\{x\right\},m^{\ast }\left\{x\right\}=0$.

4. The Lebesgue outer measure of an interval is its length.

5. If $\left\{A_n\right\}$ is a countable collection of sets of real number. Then $m^{\ast }(\cup A_n)\le m^{\ast }{A}_n$.

6. If $A$ is countable, $m^{\ast }A=0.$

7. The set $[0,1]$ is not countable.

8. Given any set $A$ and any $𝜖>0$, there is an open set $O,A\subset O$ and $m^{\ast }O\le m^{\ast }A+𝜖$. There is a $G\in G_{\delta },A\subset G$ and $m^{\ast }A=m^{\ast }G$.

1.27.3 Lebesgue Measurable Sets and Lebesgue Measure

1. Lebesgue measurable set: A set $E$ is Lebesgue measurable if for each set $A$ we have $m^{\ast }\left(A\right)=m^{\ast }(A\cap E)+m^{\ast }(A\cap A^c)$.

2. Lebesgue measure: If $E$ is said a Lebesgue measurable set, the Lebesgue measure $mE$ is the Lebesgue outer measure of $E.$

Some Important Results

1. If $m^{\ast }E=0$, then $E$ is Lebesgue measurable.

2. If $E_1$ and $E_2$ are Lebesgue measurable so $E_1\cup E_2.$

3. The family of $ℳ$ of Lebesgue measurable sets is an algebra of sets.

4. If $A$ is any set and $E_1,E_2,\dots <!--FUNCTION\; APPLICATION-->,E_n$ a finite sequence of disjoint Lebesgue measurable sets. Then $m^{\ast }(A\cap \left[\underset{i=1}{\overset{n}{\bigcup <!--FUNCTION\; APPLICATION-->}}{E}_i\right])=\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{m}^{\ast }(A\cap E_i).$

5. The collection $ℳ$ of Lebesgue measurable set is a $\sigma$-algebra.

6. Every set with Lebesgue outer measure zero is Lebesgue measurable.

7. The interval $(a,\infty )$ is Lebesgue measurable.

8. Every Borel set is Lebesgue measurable.

9. Each open set and closed set is Lebesgue measurable.

10. If $\left\{E_i\right\}$ is a sequence of Lebesgue measurable set. Then for Lebesgue measure $m(\cup E_i)\le \sum <!--FUNCTION\; APPLICATION-->m{E}_i.$ If the sets $E_i$ are pairwise disjoint then for Lebesgue measure $m(\cup E_i)=\sum <!--FUNCTION\; APPLICATION-->m{E}_i$.

11. Let $\left\{E_i\right\}$ be an infinite decreasing sequence of Lebesgue measurable sets, i.e., $E_{n+1}\subset E_n$ for each $n$. Let Lebesgue measure $mE$ is finite. Then $m\left(\underset{n=1}{\overset{\infty }{\bigcup <!--FUNCTION\; APPLICATION-->}}{E}_n\right)=\underset{n\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}m{E}_n.$

12. For a given set following are equivalent

    1. $E$ is measurable.

    2. Given $𝜖>0$, there is an open set $O\subset F$ with $m^{\ast }(O\approx F)<𝜖.$

    3. Given $𝜖>0$, there is an open set $E\subset F$ with $m^{\ast }(E\approx F)<𝜖.$

    4. $G\in G_{\delta }$ with $E\subset G,$ $m^{\ast }(G\sim E)=0.$

    5. $F\in F_{\delta }$ with $F\subset E,$ $m^{\ast }(E\sim F)=0.$ If $m^{\ast }E<\infty$, then these statements are equivalent to

    6. Given $𝜖>0$, there is a finite union $U$ of open intervals $m^{\ast }(\cup \mathrm{\Delta }E)<𝜖.$

1.27.4 Lebesgue Measurable Functions

1. Lebesgue measurable function: An extended real valued function $f$ is Lebesgue measurable if its domain is measurable and if it satisfies one of the following conditions: for each real number $\alpha$

   1. $\{x:f(x)<\alpha \}$ is measurable.

   2. $\{x:f(x)\le \alpha \}$ is measurable.

   3. $\{x:f(x)>\alpha \}$ is measurable.

   4. $\{x:f(x)\ge \alpha \}$ is measurable.

   5. $\{x:f(x)=\alpha \}$ is measurable.

2. Almost everywhere property: If a set of points where it fails to hold is set of measure zero. If $f=g$, almost everywhere if $f$ and $g$ have the same domain and $m\{x:f(x)\ne g(x\left)\right\}=0.$

3. Characteristic function ${\chi }_A$ : If $A$ is any set, the characteristic function of set $A$ is defined as ${\chi }_A=\begin{array}{c}\hfill \{\begin{array}{cc}1\hfill & \text{if}x\in A\hfill \\ 0\hfill & \text{if}x\notin A\hfill \end{array}\end{array}$

4. Simple function: A real valued function $\varphi$ is simple function, if it is Lebesgue measurable and assume only a finite number of values.

5. Borel measurability: A function $f$ is Borel measurable if for each $\alpha$, the set $\{x:f(x)>\alpha \}$ is a Borel set.

Some Important Results

1. If $f$ is an extended real valued function whose domain is measurable. Then the following statements are equivalent. For each real number $\alpha .$

   1. The set $\{x:f(x)<\alpha \}$ is Lebesgue measurable.

   2. The set $\{x:f(x)\le \alpha \}$ is Lebesgue measurable.

   3. The set $\{x:f(x)>\alpha \}$ is Lebesgue measurable.

   4. The set $\{x:f(x)\ge \alpha \}$ is Lebesgue measurable.

2. If $c$ is a constant and $f$ and $g$ two Lebesgue measurable real valued functions defined on the same domain. Then the functions $f+c,cf,f+c,g-f$ and $fg$ are also Lebesgue measurable.

3. If $\left\{f_n\right\}$ is a sequence of Lebesgue measurable functions (with the same domain of definition). Then the function $\sup <!--FUNCTION\; APPLICATION-->\{f_1,f_2,\dots <!--FUNCTION\; APPLICATION-->,f_n\},$ $\inf <!--FUNCTION\; APPLICATION-->\{f_1,f_2,\dots <!--FUNCTION\; APPLICATION-->,f_n\}$, $\sup <!--FUNCTION\; APPLICATION-->f_n,\inf <!--FUNCTION\; APPLICATION-->f_n,\overline{\lim <!--FUNCTION\; APPLICATION-->}{f}_n$ and $\underline{\lim <!--FUNCTION\; APPLICATION-->}{f}_n$ are all Lebesgue measurable.

4. If $f$ is a measurable function and $f=g$ almost everywhere, then $g$ is measurable.

5. If $f$ is Lebesgue measurable function defined on interval $[a,b]$ and assume that $f$ takes the values $\pm \infty$ only on a set of measure zero. Then given $𝜖>0$, we can find a step function $g$ and a continuous function $h$ such that $|f-g|<𝜖$ and $|f-h|<𝜖.$

1.27.5 Littlewood’s Three Principles

Littlewood’s three principles:

1. Every (measurable) set is nearly a finite union of intervals.

2. Every (measurable) set is nearly continuous.

3. Every convergent sequence of (measurable) functions is nearly uniformly convergent.

Some Important Theorems

1. If $E$ is measurable set of finite measure and $\left\{f_n\right\}$ is a sequence of measurable functions defined on $E$. If $f$ is real valued function such that for each $x\in E$, we have $f_n\left(x\right)\to f\left(x\right).$ The given $𝜖>0$ and $\delta >0$, there is a measurable set $A\subset E$ with $mA<\delta$ and an integer $N$ such that for all $x\notin A$ and all $n\ge N.$ $\left|f_n\right(x)-f(x\left)\right|<𝜖.$

2. If $E$ is measurable set of finite measure and $\left\{f_n\right\}$ a sequence of measurable functions that converge to a real valued function $f$ almost everywhere on $E$. The given $𝜖>0$ and $\delta >0$, there is a set $A\subset E$ with $ma<\delta$ and an $N$ such that for all $x\notin A$ and all $n\ge N$, $\left|f_n\right(x)-f(x\left)\right|<𝜖.$

3. Egorff’s theorem: If $\left\{f_n\right\}$ is a sequence of measurable functions that converge to a real valued function $f$ almost everywhere on a measurable set $E$ of finite measure, then given $\eta >0$, there is a subset $A\subset E$ with $mA<\eta$ such that $f_n$ converges to $f$ uniformly on $E\sim A.$

4. Lusin’s theorem: If $\left\{f_n\right\}$ is a measurable real valued function on an interval $[a,b]$. Then given $\delta >0$, there is a continuous function $\varphi$ on $[a,b]$ such that $m\{x:f(x)\ne \varphi (x\left)\right\}<\delta .$

1.28 The Lebesgue Integration

1.28.1 The Riemann Integral

If $f$ is a bounded real valued function defined on the interval $[a,b]$ and $a=x_0<\dots <!--FUNCTION\; APPLICATION--><x_n=b$ is a subdivision of $[a,b].$ $\begin{array}{ccc}\hfill S& =\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}(x_i-x_{i-1})M_{i}s\hfill & \hfill =\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}(x_i-x_{i-1})m_i\end{array}$where $\begin{array}{ccc}\hfill M_i& =\sup <!--FUNCTION\; APPLICATION-->f\left(x\right)\forall <!--FUNCTION\; APPLICATION-->x\in (x_{i-1},x_i)\text{and}{m}_i\hfill & \hfill =\inf <!--FUNCTION\; APPLICATION-->f\left(x\right)\forall <!--FUNCTION\; APPLICATION-->x\in (x_{i-1},x_i)\end{array}$The upper Riemann integral of $f$ is $R\underset{a}{\overset{-b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=\inf <!--FUNCTION\; APPLICATION-->S$. and the lower Riemann integral of $f$ is $R\underset{-a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=\sup <!--FUNCTION\; APPLICATION-->s$. The function $f$ is Riemann integrable if $R\underset{a}{\overset{-b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f=R\underset{-a}{\overset{b}{\mathrm{\int <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f=R\underset{a}{\overset{b}{\mathrm{\int <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx.$

1.28.2 Step function

For the given subdivision $a=x_0<\dots <!--FUNCTION\; APPLICATION--><x_n=b$ of the interval $[a,b],$ a function $\varphi$ is a step function if $\varphi \left(x\right)=c_i,x_{i-1}<x<x_i$.
Some Important Results

1. $\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\varphi \left(x\right)dx=\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{c}_i(x_i-x_{i-1}).$

2. $R\underset{a}{\overset{-b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=\inf <!--FUNCTION\; APPLICATION-->\underset{a}{\overset{b}{\mathrm{\int <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\varphi \left(x\right)dx.$ for all step function $\varphi \left(x\right)\ge f\left(x\right).$

3. $R\underset{-a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=\sup <!--FUNCTION\; APPLICATION-->\underset{a}{\overset{b}{\mathrm{\int <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}\varphi \left(x\right)dx.$ for all step function $\varphi \left(x\right)\le f\left(x\right).$

1.28.3 The Lebesgue Integral of a Bounded Function Over a Set of Finite Measure

1. Characteristic function: A real valued function of set $E$, ${\chi }_E=\begin{array}{c}\hfill \{\begin{array}{cc}1,\hfill & x\in E\hfill \\ 0,\hfill & x\notin E\hfill \end{array}\end{array}$

2. Simple function: A linear combination $\varphi \left(x\right)=\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{a}_i{\chi }_{E_i}\left(x\right)$.

3. Canonical representation of simple function: If $\varphi$ is a simple function and $\{a_1,\dots <!--FUNCTION\; APPLICATION-->,a_n\}$ is the set of non-zero values of $\varphi$, then $\varphi =\sum <!--FUNCTION\; APPLICATION-->a_i{\chi }_{A_i},$ where $A_i=\{x:\varphi (x)=a_i$} is called canonical representation of simple function. Here $A_i$ are disjoint and $a_i$ are distinct and non-zero.

4. Integral of simple function: If simple function $\varphi$ vanishes outside a set of finite measure, the integral of $\varphi$ is $\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->\varphi \left(x\right)dx=\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{a}_i\left(m{A}_i\right)$ where $\varphi$ has a canonical representation $\varphi =\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{a}_i{\chi }_{A_i}$ and $m{A}_i$ is the Lebesgue measure of $A_i.$ If $E$ is any measurable set, we have $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}\varphi =\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->\varphi .{\chi }_E.$

5. The Lebesgue integral: If $f$ is bounded measurable function defined on a measurable set $E$ with $mE$ is finite, the Lebesgue integral of $f$ over $E$ is $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f\left(x\right)dx=\inf <!--FUNCTION\; APPLICATION-->\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}\psi \left(x\right)dx$. for all simple function $\psi \ge f.$

Some Important Theorems

1. Let $\varphi =\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{a}_i{\chi }_{E_i}$, with $E_i\cap E_j=\varnothing$ for $i\ne j$. Suppose each set $E_i$ is a measurable set of finite measure. Then $\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->\varphi =\underset{i=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{a}_{i}m{E}_i.$

2. If $\varphi$ and $\psi$ are simple functions which vanishes outside a set of finite measure, then $\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->(a\varphi +b\psi )=a\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->\varphi +b\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->\psi$ and if $\varphi >\psi$ almost everywhere then $\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->\varphi \ge \mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->\psi .$

3. If $f$ is defined and bounded on a measurable set $E$ with $mE$ and $\underset{f\le \psi }{\inf <!--FUNCTION\; APPLICATION-->}\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}\psi \left(x\right)dx=\underset{f\le \varphi }{\sup <!--FUNCTION\; APPLICATION-->}\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}\varphi \left(x\right)dx$ for all simple functions $\varphi$ and $\psi$. Then $f$ is measurable function.

4. Let $f$ be bounded function defined on $[a,b]$. If $f$ is Riemann integrable on $[a,b]$, then it is measurable and $R\underset{a}{\overset{b}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx=\underset{a}{\overset{b}{\mathrm{\int <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f\left(x\right)dx$.

5. If $f$ and $g$ are bounded measurable functions defined on set $E$ of finite measure, then

   1. $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}(af+bg)=a\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f+b\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}g.$

   2. If $f=g$ almost everywhere, then $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f=\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}g$.

   3. If $f\le g$ almost everywhere, then $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f\le \underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}g$.

   4. $|\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->f|\le \mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->\left|f\right|.$

   5. If $A\le f\left(x\right)\le B$, then $A\left(mE\right)\le \underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f\le B\left(mE\right).$

   6. If $A$ and $B$ are disjoint measurable sets of finite measure, then $\underset{A\cup B}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f=\underset{A}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f+\underset{B}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f.$

6. Let $\left\{f_n\right\}$ be a sequence of measurable functions defined on a set $E$ of finite measure and suppose that there is a real number $M$ such that $\left|f_n\right(x\left)\right|\le M$ for all $x.$ If $f\left(x\right)=\lim <!--FUNCTION\; APPLICATION-->f_n\left(x\right)$ for each $x\in E$, then $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}=\lim <!--FUNCTION\; APPLICATION-->\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}{f}_n$.

7. A bounded function $f$ on $[a,b]$ is Riemann integrable iff the set of points at which $f$ is discontinuous has measure zero.

1.28.4 The Integral of a Non-negative Function

1. Integral of a non-negative function: if $f$ is a non-negative measurable function defined on a measurable set $E$, then integral of non-negative measurable function $f$ is $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f=\underset{h\le f}{\sup <!--FUNCTION\; APPLICATION-->}\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}h$ where $h$ is a bounded measurable function such that $m\{x:h(x)\ne 0\}<\infty .$

2. Integrable function: A non-negative measurable function $f$ is integrable over the measurable set $E$, if $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f<\infty$

Some Important Theorems

1. If $f$ and $g$ are measurable functions then

   1. $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}cf=c\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f,c>0$.

   2. $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f+g=\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f+\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}g.$

   3. If $f\le g$ almost everywhere, then $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f\le \underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}g.$

2. Fatou’s lemma: If $\left\{f_n\right\}$ is a sequence of non-negative measurable function and $f_n\left(x\right)\to f\left(x\right)$, almost everywhere on a set $E$, then $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f\le \underline{\lim <!--FUNCTION\; APPLICATION-->}\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}{f}_n$.

3. Monotone convergence theorem: If $\left\{f_n\right\}$ is an increasing sequence of non-negative measurable functions and let $f=\lim <!--FUNCTION\; APPLICATION-->f_n$ almost everywhere, then $\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->f=\lim <!--FUNCTION\; APPLICATION-->\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->f_n.$

4. If $\left\{u_n\right\}$ is a sequence of non-negative measurable functions and $f=\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}$. Then $\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->f=\underset{n=1}{\overset{\infty }{\sum <!--FUNCTION\; APPLICATION-->}}\mathrm{\int <!--nolimits-->}<!--FUNCTION\; APPLICATION-->u_n.$

5. If $F$ is a non-negative function and $\left\{E_i\right\}$ a disjoint sequence of measurable sets. Let $E=\cup E_i$. Then $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}=\sum <!--FUNCTION\; APPLICATION-->\underset{E_i}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f.$

6. Let $f$ and $g$ be two non-negative measurable functions. If $f$ is integrable over $E$ and $g\left(x\right)<f\left(x\right)$ on $E$, then $g$ is also integrable on $E$, and $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}(f-g)=\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f-\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}g.$

7. If $f$ is a non-negative function which is integrable over a set $E$. Then given $𝜖>0$ there is a $\delta >0$ such that for every set $A\subset E$ with $mA<\delta$, we have $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f<𝜖.$

1.28.5 The General Lebesgue Integral

The positive and negative par of function:
$\begin{array}{ccccc}\hfill f^{+}\left(x\right)& =\max <!--FUNCTION\; APPLICATION-->\left\{f\right(x),0\}{f}^{-}\left(x\right)\hfill & \hfill =\max <!--FUNCTION\; APPLICATION-->\{-f(x),0\}\text{and}f& =f^{+}-f^{-}\left|f\right|\hfill & \hfill =f^{+}+f^{-}\end{array}$If $f$ is a measurable function, then $f$ is integrable over $E$ if $f^{+}$ and $f^{-}$ are both integrable over $E$ and $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f\left(x\right)dx=\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}{f}^{+}\left(x\right)dx+\underset{E}{\mathrm{\int <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}{f}^{-}\left(x\right)dx$.
Some Important Theorems

1. If $f$ and $g$ are integrable over $E.$ Then

   1. The function $cf$ is integrable over $E$ and $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}cf=c\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f.$

   2. The function $f+g$ is integrable over $E$ and $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}(f+g)=\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f+\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}g.$

   3. If $f\le g$ almost everywhere, then $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f\le \underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}g.$

   4. If $A$ and $B$ are disjoint measurable sets contained in $E$, then $\underset{A\cup B}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}=\underset{A}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f+\underset{B}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f$

2. Lebesgue convergence theorem: If $g$ is integrable over $E$ and $\left\{f_n\right\}$ is a sequence of measurable functions such that $\left|f_n\right|\le g$ on $E$ and for almost all $x\in E$, we have $f\left(x\right)=\lim <!--FUNCTION\; APPLICATION-->f_n\left(x\right)$. Then $\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}f\le \lim <!--FUNCTION\; APPLICATION-->\underset{E}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}{f}_n$.

3. If $\left\{g_n\right\}$ is a sequence of integrable of measurable functions which converge almost everywhere to an integrable function $g$. If $\left\{f_n\right\}$ is a sequence of measurable functions such that $\left|f_n\right|\le g_n$ and $\left\{f_n\right\}$ converges to $f$ almost everywhere. If $\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->g=\lim <!--FUNCTION\; APPLICATION-->\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->g_n$ then $\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->f=\lim <!--FUNCTION\; APPLICATION-->\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->f_n.$

1.28.6 Convergence in Measure

1. Convergence inn measure: A sequence $\left\{f_n\right\}$ of measurable functions is said to converge to $f$ in measure if given $𝜖>0$, there is an $N$ such that for all $n\ge N$, we have $m\{x:|f\left(x\right)-f_n\left(x\right)|\ge 𝜖\}<𝜖$.

2. Cauchy sequence in measure: A sequence $\left\{f_n\right\}$ of measurable functions in Cauchy sequence in measure if given $𝜖>0$ there is $N$ such that for all $m,n\le N$, we have $m\{x:|f_n\left(x\right)-f_m\left(x\right)|\ge 𝜖\}<𝜖.$

3. Some Important Theorems

   1. If $\left\{f_n\right\}$ is a sequence of measurable functions that converges in measure to $f.$ Then there is a subsequence $\left\{f_{n_k}\right\}$ that converges to $f$ almost everywhere.

   2. If $\left\{f_n\right\}$ is a sequence of measurable functions defined on a set $E$ of finite measure. Then $\left\{f_n\right\}$ converges to $f$ in measure iff every subsequence of $\left\{f_n\right\}$ has in turn a subsequence that converges almost everywhere to $f.$

   3. Fatou’s lemma and the monotone and Lebesgue convergence theorems remain valid if “convergence almost everywhere” is replaced by “convergence in measure”.

1.29 Real Valued Functions of Several Variables

1. Real valued functions of several variable: If $D\subset {ℝ}^n$ the function $f\left(x\right)=f(x_1,x_2,\dots <!--FUNCTION\; APPLICATION-->,x_n\}$ where $x\in {ℝ}^n$ and $f:D\to {ℝ}^n$ is called real valued function of several variables.

2. Limit: A function $f\left(x\right),x\in D\subset {ℝ}^n$ has a limit $l$. i.e., $\underset{x\to a}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)=l$, if given $𝜖>0$, there exist $\delta >0$ such that $\left|f\right(x)-l|<𝜖$ for every $a\in D$, $\left|\right|x-a\left|\right|<\delta .$

3. Continuity: A function $f:D\to {ℝ}^n$ is continuous at $x=a$, if for each $𝜖>0$, there exist $\delta >0$ such that $\left|f\right(x)-f(a\left)\right|<𝜖$ whenever $\left|\right|x-a\left|\right|<\delta$ and $x\in D\subset {ℝ}^n$ or $f:D\to {ℝ}^n$ is continuous at $a$ iff $\underset{x\to a}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)=f\left(a\right).$

4. Uniformly continuity: A function $f:D\to {ℝ}^n$ is uniformly continuous on $D$ if it is continuous at every $x\in D\subset {ℝ}^n.$

5. Some Important Results

   1. $f:D\to {ℝ}^n,g:D\to {ℝ}^n$ then
      $\begin{array}{ccc}\hfill \lim <!--FUNCTION\; APPLICATION-->(f\pm g)\left(x\right)& =\lim <!--FUNCTION\; APPLICATION-->f\left(x\right)\pm \lim <!--FUNCTION\; APPLICATION-->g\left(x\right)\hfill & \hfill \\ \hfill \lim <!--FUNCTION\; APPLICATION-->(f\ast g)\left(x\right)& =\lim <!--FUNCTION\; APPLICATION-->f\left(x\right)\ast g\left(x\right)\hfill \end{array}$ If $\lim <!--FUNCTION\; APPLICATION-->g\left(x\right)\ne 0.$ $\lim <!--FUNCTION\; APPLICATION-->(f∕g)\left(x\right)=\lim <!--FUNCTION\; APPLICATION-->f\left(x\right)∕\lim <!--FUNCTION\; APPLICATION-->g\left(x\right)$

   2. The range of a function continuous on a compact set is compact.

   3. A real valued function continuous on a compact set is bounded and attains its bounds.

   4. A real valued function continuous on a closed rectangle $[a,b]$ is bounded and attains its bounds.

   5. A function continuous on a compact domain is uniformly continuous.

   6. Let $f$ be a real valued function with domain $D\subset {ℝ}^n$. Let $D$ be such that $x,y\in D$ $tx+(1+t)y\in D\forall <!--FUNCTION\; APPLICATION-->t\in [0,1]$ Then $f$ assumes every value between $f\left(x\right)$ and $f\left(y\right).$

   7. If $\underset{x\to a}{\lim <!--FUNCTION\; APPLICATION-->}f\left(x\right)=b$ and $b=(b_1,b_2,\dots <!--FUNCTION\; APPLICATION-->,b_m)$ $f=(f_1,f_2,\dots <!--FUNCTION\; APPLICATION-->,f_m)$ then $\underset{x\to a}{\lim <!--FUNCTION\; APPLICATION-->}{f}_i\left(x\right)=b_i(1\le i\le m)$ and conversely.

1.30 Partial Derivatives

1. First order: Let $S$ be an open set in Euclidean space ${ℝ}^n$ and $f:S\to ℝ$ be a real valued function on $S.$ If $x=(x_1,x_2,\dots <!--FUNCTION\; APPLICATION-->,x_n)$ and $c=(c_1,c_2,\dots <!--FUNCTION\; APPLICATION-->,c_n)$ are two point on $S$ having corresponding coordinates equal except for the $k^{th}$ if $x_i=c_i$ for $i\ne k$ and if $x_k\ne c_k$ then $D_{k}f\left(c\right)=\underset{x_k\to c_k}{\lim <!--FUNCTION\; APPLICATION-->}\frac{f\left(x\right)-f\left(c\right)}{x_k-c_k}$ is called partial derivative of $f$ with respect to $k^{th}$ coordinate.

2. Second order: If $f$ has a partial derivatives $D_{1}f,D_{2}f,\dots <!--FUNCTION\; APPLICATION-->,D_{n}f$ on an open set $S$, then these are called second order partial derivatives. The partial derivatives of $D_k$ with respect to $r^{th}$ variable is $D_{r,k}f=D_r\left(D_{k}f\right).$

3. Directional derivative: Let $f:S\to {ℝ}^n,S\subset {ℝ}^n$. The directional derivative of $f$ at $c$ in the direction $u$, denoted by the symbol $f^{\prime }(c,u)$ is $f(c,u)=\underset{h\to 0}{\lim <!--FUNCTION\; APPLICATION-->}\frac{f(c+hu)-f\left(c\right)}{h}$ whenever the limit on the right exist.

4. Total derivative: The function $f$ is said to be differentiable at $c$ if there exists a linear function $T_c:{ℝ}^n\to {ℝ}^m$ such that $T(c+v)=f\left(c\right)+T_c\left(v\right)+\left|\right|v\left|\right|E_c\left(v\right)$ where $E_0\left(v\right)\to 0$ as $v\to 0.$ Then the above equation is called first order Taylor formula and $T_c$ is a linear function is called the total derivative of $f$ at $c.$

5. Some Important Theorems

   1. Assume $f$ is differentiable at $C$ with total derivative $T_c$. Then the directional derivative $f^{\prime }(c,u)$ exists for every $u\in {ℝ}^n$ and $T_c\left(u\right)=f^{\prime }(c,u).$

   2. If $f$ is differentiable at $c$ then $f$ is continuous at $c.$

   3. Let $f:S\to {ℝ}^m$ be differentiable at an interior point $c\in S$, where $S\subseteq {ℝ}^n$.If $v=v_1{e}_1+v_2{e}_2+\cdots +v_n{c}_n$ where $e_1,e_2,\dots <!--FUNCTION\; APPLICATION-->,e_n$ are the units coordinate vectors in ${ℝ}^n,$ then $f^{\prime }\left(c\right)\left(v\right)=\underset{k=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{v}_k{D}_{k}f\left(c\right)$. In particular, if $f$ is real valued $(m=1)$, we have $f^{\prime }\left(c\right)\left(v\right)=\nabla <!--FUNCTION\; APPLICATION-->f\left(c\right)\cdot v$ The dot product of $v$ with the vector $\nabla <!--FUNCTION\; APPLICATION-->f\left(c\right)=(D_{1}f\left(c\right),D_{2}f\left(c\right),\dots <!--FUNCTION\; APPLICATION-->,D_{n}f\left(c\right)).$

   4. Let $u$ and $v$ be two real valued functions defined on a subset $S$ of the complex plane. Assume also that $u$ and $v$ differential at an interior point $c$ of $S$ and that the partial derivatives satisfy the Cauchy-Riemann equations at $c.$ Then the function $f=u+iv$ has a derivative at $c$ and also $f^{\prime }\left(c\right)=D_{1}u\left(c\right)+t{D}_{1}v\left(c\right).$

   5. Chain rule: Assume $g$ is differentiable at $a$ with total derivative $g^{\prime }\left(a\right).$ Let $b=g\left(a\right)$ and assume that $f$ is differentiable at $b$, with the total derivative $f^{\prime }\left(b\right).$ Then the composite function $h=f\circ g$ is differentiable at $a$ and the total derivative $h^{\prime }\left(a\right)$ is given by $h^{\prime }\left(a\right)=f^{\prime }\left(b\right)\circ g^{\prime }\left(a\right).$ Then the composition of the linear functions $f^{\prime }\left(b\right)$ and $g^{\prime }\left(a\right).$

   6. Let $f$ and $D_{2}f$ be continuous on a rectangle $[a,b]\times [c,d]$. Let $p$ and $q$ be differentiable on $[c,d]$, where $p\left(y\right)\in [a,b]$ for each $y\in [c,d]$. Define $F$ as $F\left(y\right)=\underset{p\left(y\right)}{\overset{q\left(y\right)}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}f(x,y)dx$ if $y\in [c,d]$ Then $F^{\prime }\left(y\right)$ exists for each $y\in (c,d)$ and $\begin{array}{ccc}\hfill F^{\prime }\left(y\right)& =\underset{p\left(y\right)}{\overset{q\left(y\right)}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}}{D}_{2}f(x,y)dx+f\left(q\right(y),y)q^{\prime }\left(y\right)\hfill & \hfill -f\left(p\right(y),y)p^{\prime }\left(y\right)\end{array}.$

   7. Mean-value theorem: Let $S$ be an open subset of ${ℝ}^n$ and assume that $f:S\to {ℝ}^n$ is differentiable at each point of $S.$ Let $x$ and $y$ be two points in $S$ such that $L(x,y)\subseteq S$. Then for every vector $a\in {ℝ}^n$ there is a point $z\in L(x,y)$ such that $a\cdot \left\{f\right(y)-f(x\left)\right\}=a\cdot \left\{f^{\prime }\right(z\left)\right(y-x\left)\right\}.$

   8. Let $S$ be an open connected subset of ${ℝ}^n$ and $f:S\to {ℝ}^m$ be differentiable at each point in $S$. If $f^{\prime }\left(c\right)=0$ for each $c\in S$, then $f$ is constant of $S.$

   9. If one of the partial derivatives $D_{1}f,D_{2}f,\dots <!--FUNCTION\; APPLICATION-->,D_{n}f$ exist at $c$ and that the remaining $(n-1)$ partial derivatives exist in some $n$-ball $B\left(c\right)$ and are continuous at $c.$ Then $f$ is differentiable at $c.$

   10. If both partial derivatives $D_{r}f$ and $D_{k}f$ exist in an $n$-ball $B(c,\delta )$ and if both are differentiable at $c$, then $D_{r,k}f\left(c\right)=D_{k,r}f\left(c\right).$

   11. If both partial derivatives $D_{r}f$ and $D_{k}f$ exists in an $n$-ball $B\left(c\right)$ and if both $D_{r,k}f$ and $D_{k,r}f$ are continuous at $c$, then $D_{r,k}f\left(c\right)=D_{k,r}f\left(c\right).$

   12. Taylor’s formula: Assume $f$ and all its partial derivatives of order $<m$ are differentiable at each point of an open set $S\in {ℝ}^n$. If $a$ and $b$ are two points of $S$, such that $L(a,b)\subseteq {ℝ}^n$, then there is a point $z$ on the line segment $L(a,b)$ such that $\begin{array}{ccc}\hfill f\left(b\right)-f\left(a\right)& =\underset{k=1}{\overset{m-1}{\sum <!--FUNCTION\; APPLICATION-->}}\frac{1}{n!}{f}^{\left(k\right)}(a,b-a)\hfill & \hfill +\frac{1}{m!}{f}^{\left(m\right)}(z,b-a)\end{array}$

1.31 Implicit Functions and Inverse Functions

1. Jacobian determinant: If $f=(f_1,f_2,\dots <!--FUNCTION\; APPLICATION-->,f_n)$ and $x=(x_1,x_2,\dots <!--FUNCTION\; APPLICATION-->,x_n)$ the Jacobian matrix is $Df\left(x\right)=\left[D_j{f}_i\right(x\left)\right]$ on $n\times n$ matrix, and Jacobian determinant is $F_f\left(x\right)=\det <!--FUNCTION\; APPLICATION-->Df\left(x\right)=\det <!--FUNCTION\; APPLICATION-->\left[D_j{f}_i\right(x).]$

2. Some Important Theorems

   1. If $f=u+iv$ is a complex-valued function with its derivative at $ℂ$, then $J_f\left(z\right)=\left|f^{\prime }\right(z){|}^2.$

   2. Let $B=B(a,r)$ be an $n$-ball in ${ℝ}^n$ and $\partial B$ denote its boundary and assume that all the partial derivatives $D_i{f}_i\left(x\right)$ exist if $x\in B$. Assume further that $f\left(x\right)\ne f\left(a\right)$ if $x\in \partial B$ and that the Jacobian determinant $J_f\left(x\right)\ne 0$ for each $x\in B$. Then $f\left(B\right)$, the image of $B$ under $f$ contains an $n$-ball with center at $f\left(a\right).$

   3. Let $A$ be an open subset of ${ℝ}^n$ and $f:A\to {ℝ}^n$ is continuous and has finite partial derivatives $D_i{f}_i$ on $A$ if $f$ is one-to-one on $A$ and if $J_f\left(x\right)\ne 0$ for each $x\in A$, then $f\left(A\right)$ is open.

   4. If $f=(f_1,f_2,\dots <!--FUNCTION\; APPLICATION-->,f_n)$ has continuous partial derivatives $D_i{f}_i$ on an open set $S$ in ${ℝ}^n$ and that the Jacobian determinant $J_f\left(a\right)\ne 0$ for some point $a\in S$. Then there is an $n$-ball $B\left(a\right)$ on which $f$ is one-to-one.

   5. Let $A$ be an open subset of ${ℝ}^n$ and assume that $f:A\to {ℝ}^n$ has continuous partial derivatives $D_i{f}_i$ on $A$. If $J_f\left(x\right)\ne 0$ for all $x\in A$, then $f$ is an open mapping.

3. Inverse function theorem: $C^{\prime }$ is a set of continuously differentiable function. If $f=(f_1,f_2,\dots <!--FUNCTION\; APPLICATION-->,f_n)\in C^{\prime }$ on an open set $S\subset {ℝ}^n$ and $T=f\left(S\right)$. If the Jacobian determinant $J_f\left(a\right)\ne 0$ for some $a\in S$. Then there are two open sets $X\subset S$ and $y\subset T$ and a uniquely determined function $g$ such that

   1. $a\in X$ and $f\left(a\right)\in Y.$

   2. $Y=f\left(X\right)$

   3. $f$ is one-to-one on $X$

   4. $g$ is defined on $Y.g\left(Y\right)=X$ and $g\left(f\right(x\left)\right)=x$ for every $x\in X$

   5. $g\in C^{\prime }$ on $Y.$

4. Implicit function theorem: Let $f=(f_1,f_2,\dots <!--FUNCTION\; APPLICATION-->,f_n)$ be a vector-valued function defined on an open set $S\subset {ℝ}^{n+k}$ with values in ${ℝ}^n$. Suppose $f\in C^{\prime }$ on $S.$ Let $(x_0,t)$ be a point in $S$ for which $f(x_0,t_0)=0$ and for which the $n\times n$ determinant $\det <!--FUNCTION\; APPLICATION-->\left[D_j{f}_i\right(x_0,t_0\left)\right]\ne 0$. Then there exist a $k$-dimensional open set $T_0$ containing $t_0$ and one and only one vector-valued function $g$ defined on $T_0$ and having values in ${ℝ}^n$, such that

   1. $g\in C^{\prime }$ on $T_0$

   2. $g\left(t_0\right)=x_0$

   3. $f\left(g\right(t),t)=0$ for every $t\in T_0.$

1.32 Extrema for Real Valued Functions

1. Stationary point and Saddle point: If $f$ is differentiable at $a$ and $\nabla <!--FUNCTION\; APPLICATION-->f\left(a\right)=0$, then the point $a$ is called stationary point. A stationary point is a saddle point if every $n$-ball $B\left(a\right)$ contains points $x$ such that $f\left(x\right)>f\left(a\right)$ and other points such that $f\left(x\right)<f\left(a\right).$

2. Some Important Theorems

   1. Second derivative test for extrema: Assume that the second order partial derivatives $D_{i,j}f$ exist on an $n$-ball $B\left(a\right)$ and are continuous at $a$, where $a$ is a stationary point of $f.$ Let $\begin{array}{ccc}\hfill Q\left(t\right)& =\frac{1}{2}{f}^{″<!--FUNCTION\; APPLICATION-->}(a,t)\hfill & \hfill =\frac{1}{2}\underset{i=1}{\sum <!--FUNCTION\; APPLICATION-->}\underset{j=1}{\overset{n}{\sum <!--FUNCTION\; APPLICATION-->}}{D}_{i,j}f\left(a\right)t_i{t}_j\end{array}$

      1. If $Q\left(t\right)>0$ for all $t\ne 0,f$ has a relative minimum at $a.$

      2. If $Q\left(t\right)<0$ for all $t\ne 0,f$ has a relative maximum at $a.$

      3. If $Q\left(t\right)$ takes both positive and negative values then $f$ has a saddle point at $a.$

   2. Let $f$ be a real-valued function with continuous second order partial derivative at a stationary point $a\in {ℝ}^2$. Let $\begin{array}{cccc}\hfill A& =D_{1,1}f\left(a\right)B\hfill & \hfill =D_{1,2}f\left(a\right)C& =D_{2,2}f\left(a\right)\hfill \end{array}$ and let $\mathrm{\Delta }=\det <!--FUNCTION\; APPLICATION-->\left[\begin{array}{ccc}\hfill A\hfill & \hfill BB\hfill & \hfill C\hfill \end{array}\right]=AC-B^2$, then

      1. If $\mathrm{\Delta }>0$ and $A>0,f$ has relative minimum at $a.$

      2. If $\mathrm{\Delta }>0$ and $A<0,f$ has relative maximum at $a.$

      3. If $\mathrm{\Delta }<0$, $f$ has a saddle point at $a$.