# Introduction to Real Analysis, 4ed, An Indian Adaptation

ISBN: 9789354244612

380 pages

## Description

Introduction to Real Analysis is a comprehensive textbook, suitable for undergraduate level students of pure and applied mathematics. Starting with the background of the notations for sets and functions and mathematical induction, the book focuses on real numbers and their properties, real sequences along with associated limit concepts, and infinite series. The book then explores the concepts of fundamental properties of limits and continuous functions, basic theory of derivatives and applications, including mean value theorem, chain rule, and inversion theorem.

- Chapter 1 Preliminaries

1.1 Sets and Functions

1.2 Mathematical Induction

1.3 Finite and Infinite Sets

Chapter 2 The Real Numbers

2.1 The Algebraic and Order Properties of ℝ

2.2 Absolute Value and the Real Line

2.3 The Completeness Property of ℝ

2.4 Applications of the Supremum Property

2.5 Intervals

Chapter 3 Real Sequences

3.1 Sequences and Their Limits

3.2 Limit Theorems

3.3 Monotone Sequences

3.4 Subsequences and the Bolzano-Weierstrass Theorem

3.5 The Cauchy Criterion

3.6 Properly Divergent Sequences

Chapter 4 Infinite Series

4.1 Introduction to Infinite Series

4.2 Absolute Convergence

4.3 Tests for Absolute Convergence

4.4 Tests for Nonabsolute Convergence

Chapter 5 Limits

5.1 Limits of Functions

5.2 Limit Theorems

5.3 Some Extensions of the Limit Concept

Chapter 6 Continuous Functions

6.1 Continuous Functions

6.2 Combinations of Continuous Functions

6.3 Continuous Functions on Intervals

6.4 Uniform Continuity

6.5 Continuity and Gauges

6.6 Monotone and Inverse Functions

Chapter 7 Differentiation

7.1 The Derivative

7.2 The Mean Value Theorem

7.3 L’Hospital’s Rules 1

7.4 Taylor’s Theorem

Chapter 8 The Riemann Integral

8.1 Riemann Integral

8.2 Riemann Integrable Functions

8.3 The Fundamental Theorem

8.4 The Darboux Integral

Chapter 9 Sequences and Series Of Functions

9.1 Pointwise and Uniform Convergence

9.2 Interchange of Limits

9.3 Series of Functions

9.4 The Exponential and Logarithmic Functions

9.5 The Trigonometric Functions

Chapter 10 The Generalized Riemann Integral

10.1 Definition and Main Properties

10.2 Improper and Lebesgue Integrals

10.3 Infinite Intervals

10.4 Convergence Theorems

Chapter 11 A Glimpse into Topology

11.1 Open and Closed Sets in ℝ

11.2 Compact Sets

11.3 Continuous Functions

11.4 Metric Spaces

Chapter 12 Functions of Several Real Variables

Appendix A Logic and Proofs

Appendix B The Riemann And Lebesgue Criteria

Appendix C Two Examples

Appendix D Multiple Choice Questions

References

Photo Credits

Hints for Selected Exercises

Index