MATH-305: Real Analysis-I

This is the first rigorous course in analysis and has a theoretical emphasis. It tegorously develops the fundamental ideas of calculus and is aimed to develop the students’ ability to deal with abstract mathematics and mathematical proofs.

Real Number System: Ordered fields, The field of Reals, The extended real number system, Euclidean space. Numerical Sequences and Series: Limit of a sequence, Bounded sequence, Monotone sequences, Limit superior and inferior, Subsequences, Infinite series of constants, Test for convergence of series, Absolute and conditional convergence. Continuity: Limit of a function and continuous function, Continuity and compactness, Continuity and connectedness, Uniform continuity, Kind of discontinuities. Differentiation: The derivative of a real function, Mean value theorems, The continuity of derivatives, Taylor’s theorem. Riemann Stieltjes Integral: Definition of Riemann Integral, Upper and lower sums, Integrability criterion, Classes of integrable functions, Properties of the Riemann Integral.

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4. Rabenstein R.L.,1984. Elements of Ordinary differential equations. 1st ed. Academic Press.
5. Bartle R,G, Donald R.S., 1999. Introduction to Real Analysis, 3rd ed. Wiley.
6. Royden H. 1988. Real Analysis. 3rd ed. Prentice Hall/ Pearson Edition.