#### 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 |