Course Title: 
Real Analysis
Course Code: 
Year Taught: 
Integrated Degree
School of Arts and Sciences

'Real Analysis' is a course offered in Fifth Semester of B. Sc. - B. Ed. in Physics, Mathematics, Computer Science program at the School of Arts and Sciences, Amrita Vishwa Vidyapeetham, Mysuru campus.

On completion of the course, the student teacher will;

To enable students to understand the basic properties of the field of real numbers and understand notion of continuous functions and their properties.

Unit I: 

Review: Sets and Functions, Mathematical Induction, Finite and Infinite Sets.

The Real Numbers:The Algebraic and Order Properties of R, Absolute Value and the Real Line, The Completeness Property of R, Applications of the Supremum Property.
Chapter-2 (Sec.2.1-2.3).

Unit II: 

Sequences and Series-Sequences and Their Limits, Limit Theorems, Monotone Sequences, Subsequences and the Bolzano-Weierstrass Theorem, The Cauchy Criterion, Properly Divergent Sequences, Introduction to Infinite Series.
Chapter-3 (Sec.3.1-2.6).

Review of Limits-Limits of Functions, Limit Theorems. (Chapter-4, review only)

Unit III:

Continuous Functions-Continuous Functions, Combinations of Continuous Functions,Continuous Functions on Intervals, Uniform Continuity.
Chapter-5 (Sec.5.1-5.4)

Unit IV:

Differentiation-The Derivative, The Mean Value Theorem, L'Hospital's Rules, Taylor's Theorem.
Chapter-6 (Sec.6.1-6.4)

Unit V:

The Riemann Integral- Riemann Integral, Riemann Integrable Functions, The Fundamental Theorem.
Chapter-7 (Sec.7.1-7.4)

  1. Introduction to Real Analysis, by Robert Gardner Bartle, Donald R. Sherbert, Fourth Edition, John Wiley and Sons, 2011.
  1. Mathematical Analysis by Tom M. Apostol, Second Edition, Narosa publishing house, New Delhi,1989.
  2. Principles of Mathematical Analysis by Rudin.W, Third Edition, McGraw-Hill International Editions, 1976.
  3. Real Analysis by H.L. Royden and P.M.Fitzpatrick.Fourth Edition. Pearson Education Asia Limited, 2010.