Real Analysis

Paper Code: 
CMAT 311
Credits: 
4
Contact Hours: 
60.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Develop an understanding of real numbers, limit points, open and closed sets. 
  2. An introduction to limit and convergence of a sequence, continuous functions on closed intervals.
  3. Riemannian integration and proper integrals.

Course Outcomes (COs):

 Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

CMAT 311

 

 

 

 

 

 

 

Real Analysis

 (Theory)

 

 

 

 

 

 

The students will be able to –

 

CO31: Describe fundamental properties of the real numbers that lead to the formal development of real analysis.

CO32: Demonstrate an understanding of limits and convergence of sequences.

CO33: Understand the concept of continuous functions on closed interval and derivable functions .

CO34: Demonstrate the ability to integrate knowledge and ideas of Riemannian integration.

CO35: Analyse the convergence of Improper integrals and solve related problems.

CO36: Construct rigorous mathematical proofs of basic results in real Analysis•

Approach in teaching:

 

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Giving tasks

 

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

Unit I: 
I
9.00
Real number system as a complete ordered field, Open and closed sets, Limit point of sets, Bolzano Weirstrass theorem, Concept of compactness, Heine Borel theorem. 
 
Unit II: 
II
9.00

Real sequences, Limit and convergence of a sequence, Monotonic sequences, Cauchy’s sequences, Sub sequences and Cauchy’s general principle of convergence.

Unit III: 
III
9.00

Properties of continuous functions on a closed interval, Derivable functions: Derivative of composite function, Inverse function theorem, Limit and continuity of a function of two variables, Rolle’s and Darboux theorem.

Unit IV: 
IV
9.00

Riemann Integration, Lower and upper Riemann integral, Properties of Riemann integration, Mean value theorem of integral calculus, Fundamental theorem of integral calculus.

Unit V: 
V
9.00

Improper integrals: Kinds of improper integral, Tests of convergence of improper integrals and related problems.

Essential Readings: 
  • Shanti Narayan,  A Course of Mathematical Analysis, S. Chand and Co., New Delhi, 2005.
  • T. M. Apostol, Mathematical Analysis, Norosa Publishing House, New Delhi, 2002.
  • K. C. Sarangi, Real Analysis and Metric Spaces, Ramesh Book Depot, Jaipur, 2017.
  • Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis, John Wiley & Sons Canada, 2011.
References: 
  • P. K. Jain and S.K. Kaushik, An Introduction to Real Analysis, S.Chand and Co., New Delhi, 2000.
  • S. Lang, Undergraduate Analysis, Springer-Verlag, 2005.
  • R.R. Goldberg, Real Analysis, Oxford and IBH Publishing Company, New Delhi, 2020.
  • Charles Chapman Pugh, Real Mathematical Analysis, Springer, 2010.
  • Stephen Abbott, Understanding Analysis, Springer, 2010.
Academic Year: 
2023-2024
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