Integrated MSc-PhD Program


KSM1C02:Analysis I


Metric spaces, Limits of Functions, Continuous Functions, Continuity and Compactness, Continuity and Connectedness, Discontinuities, Monotonic Functions, Infinite Limits and Limits at Infinity, The Derivative of a Real Function, Mean Value Theorems, The Continuity of Derivatives, L’Hospital’s Rule, Derivatives of Higher Order, Taylor’s Theorem, Differentiation of Vector-valued Functions, Definition and Existence of the Integral, Properties of the Integral, Integration and Differentiation, Integration of Vector-valued Functions, Sequences and Series of Functions, Uniform Convergence, Equicontinuous Families of Functions, The Stone-Weierstrass Theorem, Functions of Several Variables, Differentiation, The Contraction Principle, The Inverse Function Theorem, The Implicit Function Theorem, The Rank Theorem.

Suggested texts:

            1. Walter Rudin, Principles of Mathematical Analysis, McGraw Hill Education; Third edition (Chapters 3,4,5,6,7,9)
            2. Terence Tao, Analysis. I & II, third ed., Texts and Readings in Mathematics, vol. 37, Hindustan Book Agency, New Delhi.
            3. Tom M. Apostol, Mathematical Analysis.