Countability of a set, Countability of rational numbers, Uncountability of real numbers. Limit point of a set, Bolzano-Weirstrass theorem, Open sets, Closed sets, Dense sets. Subsequence, Limit superior and limit inferior of a sequence, Cauchy criterion for convergence of a sequence, Monotone subsequence. Tests of convergence of series, Abel’s and Dirichlet’s tests for series, Riemann rearrangement theorem. Continuous functions on closed and bounded intervals, Intermediate value theorem, Monotone functions, Continuous monotone functions and their invertibility, Discontinuity of monotone functions. Uniform continuity, Equivalence of continuity and uniform continuity on closed and bounded intervals, Lipschitz condition, Other sufficient condition for uniform continuity. Riemann Integration: Darboux’s integral, Riemann sums and their properties, Algebra of Riemann integrable functions, Class of Riemann integrable functions, Mean value theorem, Fundamental theorems of calculus, Change of variable formula (statement only), Riemann-Stieltjes integration (definition). Taylor’s theorem and Taylor’s series, Elementary functions. Improper integral, Beta and Gamma functions.
- R. G. Bartle, D. R. Sherbert, “Introduction to Real Analysis”, John Wiley & Sons, 1992.
- K. A. Ross, “Elementary Analysis”, Undergraduate Texts in Mathematics, Springer, 2013.
- T. M. Apostol, “Calculus Vol. I”, Wiley-India edition, 2009.
- S. K. Berberian, “A First Course in Real Analysis”, Undergraduate Texts in Mathematics, Springer-Verlag, 1994.