IIT JAM EXAM – REAL ANALYSIS NOTES (IIT JAM 2023)

The notes and Solved Examples for IIT JAM Mathematics have been prepared according to the Mathematics exam syllabus.

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Table of Content Sequence Series Function Continuity Differentiability Reimann Integration Uniformly Convergence Function of several variables

SEQUENCE

A sequence of real number is a function ‘f’ whose domain is the set N of all natural numbers and range is a subset of R.

A sequence is usually denoted by {a_n} or <a_n> where f(n) = a_n, a_n is called n^th term of the sequence.

Some Important Example of Sequence :

Cluster Point (or limit point) of a sequence:

Limit of a sequence:

Convergent Sequence:

A sequence is called a convergent sequence if limit of this sequence exist.

Example:

Divergent Sequence:

A monotonic and unbounded sequence is called a divergent sequence.

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Series

Convergence of Geometric series:

Test for convergence :

Continuity

Uniform continuity

Function :

Let A ⊆ R and B ⊆ R, then a rule in which assign every element of A to unique element of B is called a function from A to B and denoted by f : A → B, where A is called a domain and B is called a co-domain.

One-one function :

A function is called a one-one function if image of all distinct element are distinct. i.e., If f : A→B and  f(x) is one –one then x_1≠ x_{2 ⇒} f(x₁) _≠ f(x₂) for every x₁, x₂ ∈ A.

Onto-function :

A function is called a onto if range set is equal to co-domain.

Limit of function :

Let  S ⊆R and α ∈ S`.

Let f : S →R, we say l∈ R is a limit of f.

⇔       for any ε> 0, ∃δ> 0 such that x₁, x₂ ∈{x : 0 < |x – α| <δ}⇒       |f(x₁) – f(x₂)| <ε

One-side limit :

Note : Limit of f(x) exist at x = a, iff RHL = LHL at x = a.

Differentiablity

Derivative of a function at a point:

Examples:

Riemann Integration

UNIFORMLY CONVERGENCE

Sequence of functions:

Example:

Point wise Limit :

Point wise convergence:

Example:

Uniformly convergent :

M_n test for uniform convergence:

Example:

Series of functions:

Results on uniform convergence of series:

(1)Weierstrass’s M test:

FUNCTION OF SEVERAL VARIABLES

Limit of a function of two variables:

Example

Continuity of a function at a point:

Example

Directional Derivative:

Differentiability of function of several variables:

 

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