Utkal University, Odisha, Mathematics Semester 2, Real Analysis Syllabus

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Unit - 1 Review Of Algebraic And Order Properties Of R

Unit 1

Review of Algebraic and Order Properties of R

1.1 Review of Algebraic and Order Properties of R

1.2 εneighborhood of a point in R

1.3 Bounded above sets Bounded below sets Bounded Sets Unbounded sets

1.4 Suprema and Infima the Completeness Property of R the Archimedean Property

1.5 Density of Rational and Irrational numbers in R.

1.6 Intervals Interior point Open Sets Closed sets Limit points of a set

1.7 Illustrations of BolzanoWeierstrass theorem for sets

1.8 Closure interior and boundary of a set

Unit - 2 Sequences And Subsequences

Unit 2

Sequences and Subsequences

2.1 Sequences and Subsequences

2.2 Bounded sequence Convergent sequence Limit of a sequence

2.3 Limit Theorems Monotone Sequences

2.4 Divergence Criteria Bolzano Weierstrass theorem for Sequences

2.5 Cauchy sequence Cauchy’s Convergence Criterion

2.6 Infinite series convergence and divergence of infinite series

2.7 Cauchy Criterion

2.8 Tests for convergence Comparison test

2.9 Limit Comparison test Ratio Test

2.10 Cauchy’s nth root test

2.11 Integral test

2.12 Alternating series Leibniz test

2.13 Absolute and Conditional convergence

Unit - 3 Limits Of Functions

Unit 3

Limits of functions

3.1 Limits of functions epsilondelta approach

3.2 Sequential criterion for limits divergence criteria

3.3 Limit theorems one sided limits Infinite limits and limits at infinity

3.4 Continuous functions sequential criterion for continuity discontinuity

3.5 Algebra of continuous functions Continuous functions on an interval

3.6 Boundedness Theorem Maximum Minimum Theorem

3.7 Bolzano’s Intermediate value theorem Location of roots theorem preservation of intervals theorem

3.8 Uniform continuity nonuniform continuity criteria Uniform continuity theorem

3.9 Monotone and Inverse Functions

Unit - 4 Differentiability Of A Function

Unit 4

Differentiability of a function at a point in an interval

4.1 Differentiability of a function at a point in an interval

4.2 Caratheodorys theorem chain Rule

4.3 Algebra of differentiable functions Mean value theorem

4.4 Interior extremum theorem Rolles theorem

4.5 Intermediate value property of derivatives Darbouxs theorem

4.6 Applications of mean value theorem to inequalities

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Other Subjects of Semester-2
Differential equations
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