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.