B.Sc. Mathematics
4th Semester Syllabus
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MJC-05 : Theory of Real Functions
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Unit 1: Limits of Functions
◦Limit of functions
◦Sequential criterion for limits
◦Divergence criteria
◦Limit theorems
◦One-sided limits
◦Infinite limits and limits at infinity
Unit 2: Continuity
◦Continuous functions
◦Sequential criterion for continuity and discontinuity
◦Algebra of continuous functions
◦Properties of continuous functions on closed intervals
◦Uniform continuity
◦Uniform continuity theorem
Unit 3: Differentiability
◦Differentiability of a function
◦Algebra of differentiable functions
◦Increasing and Decreasing functions
◦Sign of derivatives
◦Chain rule
◦Darboux's theorem
◦Intermediate value theorem for derivatives
◦Rolle's theorem
◦Lagrange's and Cauchy's Mean value theorem and their applications
Unit 4: Taylor's Theorem
◦Taylor's theorem with Lagrange's and Cauchy's remainder forms
◦Maclaurin's theorem
◦Application of Taylor's theorem in error estimation
◦Extreme values
◦Theorems related to extrema
Reference Books:
- Bartle, Robert G., & Sherbert, Donald R. (2015). Introduction to Real Analysis (4th ed.). Wiley India Edition. New Delhi
- Ross, Kenneth A. (2013). Elementary Analysis: The theory of calculus (2nd ed.). Undergraduate Texts in Mathematics, Springer. Indian Reprint
- Malik, S. C. & Arora, S. (2021). Mathematical Analysis (6th ed.). New Age International Publishers, New Delhi
- Jha,K.K. Advanced Course in Real Analysis and Higher Analysis. New Bharat Publishing House