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Title Real Analysis (Code-09050501)

 S.N Content of the topics Learning Objectives Teaching Guidelines Methodology Time (hours) 1 ·        Riemann integral, integrability of continuous and monotonic function, ·        The Fundamental theorem of integral calculus, ·        Mean value theorems of integral calculus, Student will be able to understand the study of Riemann integral, Improper integral and their convergence Lecture should be effective so that student will be able to grasp the topics easily Assignments/ Seminars/ Class tests/ Presentations 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. Student will be able to understand the study of Metric space Lecture should be effective so that student will be able to grasp the topics easily Assignments/ Seminars/ Class tests/ Presentations 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. Student will be able to understand the study of Compactness and connectedness Lecture should be effective so that student will be able to grasp the topics easily Assignments/ Seminars/ Class tests/ Presentations 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 Student will be able to understand the study of Compactness and connectedness Lecture should be effective so that student will be able to grasp the topics easily Assignments/ Seminars/ Class tests/ Presentations 15