Basic Real Analysis

By Prof. I. K. Rana   |   IIT Bombay
Learners enrolled: 2677
This s first course in Real Analysis. The aim of the course is to over the basic concepts like Real line, Topological concepts of real line, differentiation and integration with applications. Introduction to calculus of several variables.

INTENDED AUDIENCE : Any discipline, with proper exposure to Calculus.
PREREQUISITES : Basic Calculus exposure
Course Status : Completed
Course Type : Core
Duration : 12 weeks
Category :
  • Mathematics
Credit Points : 3
Level : Undergraduate
Start Date : 18 Jan 2021
End Date : 09 Apr 2021
Enrollment Ends : 01 Feb 2021
Exam Date : 25 Apr 2021 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1 : Review of sequences and series of real numbers.
Week 2 : Tests for convergence of Series. Limit superior and limit inferior.
Week 3 : Cauchy sequences and completeness of R.
Week 4 : Basic notions of Metric Spaces with emphasis on Rn. Connectedness, Compactness, and Heine Borel Theorem.
Week 5 : Continuity and Uniform continuity.
Week 6 : Monotone functions and functions of bounded variation.
Week 7 : Derivatives. Mean Value Theorem and applications
Week 8 : Riemann Stieltjes integral. Riemann`s Criterion for integrability. Improper integrals and the Gamma function.
Week 9 : Sequences and series of functions. Uniform convergence.
Week 10 : Functions of several variables: Directional derivative, partial derivative, total derivative,
Week 11 : Mean Value Theorem, Taylor`s Theorem and applications to Maxima/Minima and convexity.
Week 12 : Double and triple integrals. Statement of Fubini's Theorem and change of variable formula (without proofs) with illustrations.

S. Venkitesh

S. Venkitesh is a final year Ph.D. student at IIT Bombay, with research interests in Combinatorics and Theoretical Computer Science, working with Prof. Srikanth Srinivasan. He has co-authored 3 research papers till date. His current research focuses on problems in Boolean function and circuit  complexity,  and theory and algorithms for linear codes, which have an inherent combinatorial/algebraic flavour. Two key methods often employed in such research are the polynomial method and the probabilistic method.
He obtained his Masters’ degree in Mathematics from IIT Bombay (with Third rank) and Bachelors’ degree  in Mathematics from University of Mumbai (with First rank). He has so far been a Teaching Assistant in 7 Mathematics courses of NPTEL and 10 courses at the Department of Mathematics,  IIT Bombay.  He  also frequently accompanies the faculty of the Department of Mathematics, as a resource person for mathematics workshops at colleges in Mumbai.

Books and references

1) T. M. Apostol, Mathematical Analysis, 2nd ed., Narosa Publishers, New Delhi, 2002.
2) R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, 4th ed., John Wiley, New York, 2011.
3) R. R. Goldberg, Methods of Real Analysis, 4th Edition, Oxford and IBH Publishing Co., New Delhi.
4) K. A. Ross, Elementary Analysis: The Theory of Calculus, 2nd ed., Springer (India), New Delhi, 2013.

Instructor bio

Prof. I. K. Rana

IIT Bombay
I am Emeritus Fellow at Department of Mathematics, IIT Bombay. I have taught couses at B.Tech, M.Sc., and Ph.D level for the last 34 years.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 25 April 2021 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.


Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Bombay.It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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