Introduction to real analysis An Indian adaptation
Robert G Bartle, Donald R Sherbert
- 4
- New Delhi Wiley 2023
- 361P
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 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
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.