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Title Real Analysis (Code 09010501)

S.N Unit Content Domain Hours as per UGC
1 one 1.        Riemann integral, integrability of continuous and monotonic function,

2.       The Fundamental theorem of integral calculus,

3.       Mean value theorems of integral calculus,

4.       Improper integral and their convergence,

5.       Comparison test, Abel’s test and Dirichlet’s test,

6.       Frullani’s integrals

7.        Integral as a function of parameters

8.       Continuity, Differentiablity and integrability of an integral of a function of parameter.


Must Know 15
2 Two 1.        Definition and examples of Metric spaces,

2.       Neighborhoods, limit points, Interior points,

3.       Open and closed sets, closure and interior, boundary points,

4.       Subspace of a metric space, equivalent metrics,

5.       Cauchy sequences, Completeness,

6.       Cantor’s intersection theorem,

7.        Baire’s Category theorem, contraction principle.

Must know 15
3 Three 1.        Continuous functions, uniform continuity,

2.       Compactness for metric spaces,

3.       Sequential compactness, Bolzano- Weierstrass property,

4.       Total boundedness, Finite intersection property,

5.       Continuity in a relation with compactness, connectedness, components, continuity in relation with connectedness

Must know


Nice to know



Books Recommended;

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