Math-IVA (Analysis)

Course Code: MM 253 | Credits: 4 | Prerequisite: First year Mathematics courses

Unit I:(Sequences) Rational numbers and real numbers; sequences and subsequences; monotonicity, boundedness and convergence; the limit of a sequence, the Cauchy criterion and the Bolzano-Weierstrass theorem.
Unit II:(Continuity) The limit of a function; continuity in both the sequential and the ε-δ definitions; examples of continuous functions; the intermediate value theorem.
Unit III:(Differentiation) Differentiation and the chain rule; the mean value theorems and their applications; Taylor’s theorem; L’Hospital’s rule; maxima and minima problems.
Unit IV:(Integration) Integration in Riemann’s original definition, and the statement of the fundamental theorem of calculus.
Unit V:(Infinite Series) Rigorous definition of the convergence of an infinite series, and the application of integration to the summability of series.

Explain the basics of sequences and their various types, work in the ε-δ language, and prove basic theorems.
Explain the concept of a continuous function with examples and basic theorems.
Explain differentiability at a point, with examples and basic theorems.
Explain integrable functions with examples and theorems, and state the fundamental theorem of calculus.
Apply sequences and integration to study infinite series rigorously.

Bartle, Robert G. and Sherbert, Donald R. Introduction to Real Analysis. Wiley, 2011
Kumar, Ajit and Kumaresan, S. A Basic Course in Real Analysis. Chapman and Hall, 2014
Goldberg, Richard R. Methods of Real Analysis. Wiley, 1976
Apostol, Tom M. Mathematical Analysis. Pearson, 1974