Real Analysis : CSIR NET JRF Syllabus

Elementary Set Theory

• Sets and subsets
• Operations on sets: union, intersection, difference, complement
• Cartesian product
• Relations and functions

Finite, Countable, and Uncountable Sets

• Cardinality of sets
• Countable and uncountable sets
• Cantor’s diagonal argument

Real Number System

• Construction of real numbers
• Properties of real numbers
• Completeness of the real number system
• Archimedean property
• Supremum and infimum

Sequences and Series

• Definition of sequences and series
• Convergence and divergence of sequences and series
• Cauchy sequences
• Limsup and liminf
• Tests for convergence of series

Bolzano-Weierstrass Theorem

• Statement and proof of the theorem
• Applications of the theorem

Heine-Borel Theorem

• Statement and proof of the theorem
• Applications of the theorem

Continuity

• Definition of continuity
• Properties of continuous functions
• Intermediate value theorem
• Uniform continuity
• Lipschitz continuity

Differentiability

• Definition of differentiability
• Properties of differentiable functions
• Mean value theorem
• Taylor’s theorem
• Implicit function theorem
• Inverse function theorem

Sequences and Series of Functions

• Pointwise and uniform convergence
• Weierstrass M-test
• Uniform convergence and continuity
• Uniform convergence and integrability

Riemann Sums and Riemann Integral

• Definition of Riemann sums and Riemann integral
• Properties of Riemann integrable functions
• Improper integrals

Monotonic Functions and Discontinuities

• Definition of monotonic functions
• Types of discontinuities
• Functions of bounded variation
• Lebesgue measure and integral

Functions of Several Variables

• Partial derivatives
• Directional derivatives
• Chain rule
• Implicit function theorem

Metric Spaces

• Definition of metric spaces
• Open and closed sets
• Compactness
• Connectedness

Normed Linear Spaces

• Definition of normed linear spaces
• Examples of normed linear spaces
• Banach spaces and completeness

Spaces of Continuous Functions

• Definition of space of continuous functions
• Uniform convergence and continuity
• Arzela-Ascoli theorem