Practice Questions for CSIR NET/GATE, IIT JAM : Real Analysis
Practice multiple-choice questions (MCQs) on Real Analysis for CSIR NET JRF Mathematics,
IIT JAM, and IIT GATE exams to help candidates in successfully qualifying.
1. Countability of Set
3. Sequences of Real Numbers
4. Series of Real Numbers
6. Functions and Their Properties
7. Continuity
8. Differentiability
9. Uniform Convergence
10. Function of Several Variables
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
Explore More Resources
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