Syllabus
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Course Code: B-MAT 601 Course Name: Real Analysis -II |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Riemann integral, Integrabililty of continuous and monotonic functions, The Fundamental theorem of integral calculus. Mean value theorems of integral calculus. | |
| 2 | Improper integrals and their convergence, Comparison tests, Abel’s and Dirichlet’s tests, Frullani’s integral, Integral as a function of a parameter. Continuity, Differentiability and integrability of an integral of a function of a parameter. | |
| 3 | Definition and examples of metric spaces, Subspace of a metric space. Open spheres and closed spheres, Neighbourhoods, Open sets, Interior, exterior and boundary points, Limit points and isolated points, Closed sets, Interior and closure of a set, Boundary of a set, Bounded sets, Distance between two sets, Diameter of a set. | |
| 4 | Convergent and Cauchy sequences, Completeness of metric spaces, Cantor’s intersection theorem. Dense sets and separable spaces, Nowhere dense sets and Baire’s category theorem, Continuous and uniformly continuous functions, Homeomorphism, Banach contraction principle. | |
