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#### Real Analysis (Code-09050501)B.Sc. (Hons) – 5th semesterSYLLABUS

 S.No. Topic Domain Hours as per UGC 1 Riemann integral, integrability of continuous and monotonic function, The Fundamental theorem of integral calculus, Mean value theorems of integral calculus, Must know 15 2 Improper integral and their convergence, Comparison test, Abel’s test and Dirichlet’s test, Frullani’s integrals Integral as a function of parameters Continuity, Differentiablity and integrability of an integral of a function of parameter. Must know 15 3 Definition and examples of Metric spaces, Neighborhoods, limit points, Interior points, Open and closed sets, closure and interior, boundary points, Subspace of a metric space, equivalent metrics, Cauchy sequences, Completeness, Cantor’s intersection theorem, Baire’s Category theorem, contraction principle. Must know 15 4 Continuous functions, uniform continuity, Compactness for metric spaces, Sequential compactness, Bolzano- Weierstrass property, Total boundedness, Finite intersection property, Continuity in a relation with compactness, connectedness, components, continuity in relation with connectedness Nice to know 15

Books Recommended;

•  P.K. Jain and Khalil Ahmed: Metric spaces, 2nd Ed., Narosa, 2004
• T.M. Apostol: Mathematical Analysis, Narosa Publishing House, New Delhi, 1985
• R.R. Goldberg : Real Analysis Oxford & IBH publishing Co., New Delhi, 1970
• D. Somasundaram and B. Choudhary : A First Course in Mathematicl Analysis, Narosa Publishing House,  New Delhi, 1997
• Shanti Narayan : A Course of Mathematical Analysis, S. Chand & Co., New Delhi