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Course Co-ordinated by IIT Kharagpur
 Coordinators IIT Kharagpur

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This is a basic course in Real Analysis which is a back bone of any course on pure & applied Mathematics and Statistics. This is a very useful course for any branch of science and engineering. The present course has been designed to introduce the subject to undergraduate/postgraduate students in science and engineering. The course contains a good introduction to each topic and an advance treatment of theory at a fairly understandable level to the students at this stage. Each concept has been explained through examples and application oriented problems

 Week Topics 1 Finite, countable & uncountable sets -3 lectures. Metric Space, Open set, Closed set, Limit point, Closure of a set -3 lectures 2 Ordered set, least upper bound, greatest lower bound -2 lectures Open cover, compact set & some properties of compact set 2 lectures, Heine-Borel theorem 1 lecture. Weierstrass Theorem, connected Set- 1 lecture 3 Limit of sequences of real number 2 lectures. Some important limits, Ratio test, Cauchy limit theorem -4 lectures 4 Some theorems on limit & Bolzano Weierstrass theorem - 2 lectures Theorems on convergent & divergent sequences- 2 lectures. Cauchy sequence & its properties- 2 lectures 5 Infinite series of real numbers 2 lectures, Comparison test for series, Absolutely convergent and conditional convergent series 2 lectures. Some test for convergence of series-2 lectures 6 Raabe Test & its application 1 lecture. Limit of functions, cluster point 3 lectures 7 Limit theorems for functions 4 lectures 8 Continuity of functions 2 lectures. Properties of continuous functions & composition of continuous functions 2 lectures 9 Boundedness theorem, Bolzano theorem 2 lectures. Uniform & absolute continuity-2lectures. Type of discontinuities 2 lectures 10 Differentiability of functions of real variables 2 lectures. Mean value theorems 2 lectures. Application of derivatives , mean value theorems & Darboux theorem 2 lectures 11 LHospital rule, indeterminate forms, Taylor theorem 4 lectures. Riemann, Riemann-Stieltjes integral 2 lectures 12 Existance of Riemann, Riemann-Stieltjes integral 2 lectures Prperties of Riemann-Stieltjes integral - 2 lectures. Classes of Riemann integrable functions, monotonic functions, step functions 2 lectures

1. Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert
2. Real Analysis by H.L. Royden.
3. A First Course in Real Analysis by M.H. Protter and C.B. Morrey.
4. Principles of mathematical analysis by Walter Rudin

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