References
[1] Principles of Mathematical Analysis. Walter Rudin, 3\(^{\text {rd}}\) Edition, 1976. McGraw-Hill, Inc. ISBN 0-054234-X
[2] S. L. Gupta and Rani Nisha, 4\(^{\text {th}}\) Ed. 2004, Fundamental Real Analysis, N. Vikas.(India). ISBN 9788125909897
[3] R. G. Bartle, The Elements of Real Analysis, 3\(^{\text {rd}}\) Ed. 1982, J.Wiley. ISBN 9780471063919.
[4] Robert G bartle, Donald R Sherbet, Introduction to Real Analysis, 3\(^{\text {rd}}\) Ed. 2000, J. Wiley. ISBN 9780471321484.
Pre-requisites: Introduction to Real Analysis
Course content
- 1.
- Topology of the Real Line Open and closed subsets of \(\mathbb {R}\): neighbourhoods, interior points, exterior and boundary points, limit points of a set: Bolzano-Weistrass theorem: Cantor intersection theorem; compactness, the Heine-Borel theorem; compact sets and connected sets.
- 2.
- Continuous functions
Continuity at a point and on a set; properties of continuous functions on closed intervals; uniform continuity; the intermediate value theorem; preservation of connectedness and of compactness under continuous mappings; the maximum and minimum value theorem; uniform convergence; step function approximation; functions of bounded variation.
- 3.
- Functions of bounded variation Algebraic properties of functions of bounded variation; functions of bounded variation as a difference of two increasing functions; continuity of functions of bounded variation; absolute continuity of functions of bounded variation.
- 4.
- Differentiable functions Differentiability at a point and on a set; differentiability and continuity; some theorems on differentiation; Rolle’s theorem; Mean-value and generalized mean value theorem with applications; Taylor’s theorem with applications.
- 5.
- The Riemann integral Partition of an interval; refinements of a partition; Riemann sums; Riemann integral and the integrability criterion of Riemann classes of integrable functions; properties of the Riemann integral; integral Calculus; the first mean value theorem for integrals; the Riemann Stieljes integral; improper integrals.