Introduction to Mathematical Analysis

Abstract

Chapter 1: Tools for Analysis. 1.1 Basic Concepts of Set Theory. 1.2 Functions. 1.3 The Natural Numbers and Mathematical Induction. 1.4 Order Field Axioms. 1.5 The Completeness Axiom for the Real Numbers. 1.6 Applications of the Completeness Axiom. Chapter 2: Sequences. 2.1 Convergence. 2.2 Limit Theorems. 2.3 Monotone Sequences. 2.4 The Bolzano-Weierstrass Theorem. 2.5 Limit Superior and Limit Inferior. 2.6 Open Sets, Closed Sets, and Limit Points. Chapter 3: Limits and Continuity. 3.1 Limits of Functions. 3.2 Limit Theorems. 3.3 Continuity. 3.4 Properties of Continuous Functions. 3.5 Uniform Continuity. 3.6 Lower Semicontinuity and Upper Semicontinuity. Chapter 4: Differentiation. 4.1 Definition and Basic Properties of the Derivative. 4.2 The Mean Value of Theorem. 4.3 Some Applications of the Mean Value Theorem. 4.4 L’Hospital’s Rule. 4.5 Taylor’s Theorem. 4.6 Convex Functions and Derivatives. 4.7 Nondifferentiable Convex Functions and Subdifferentials. Chapter 5: Solutions and Hints for Selected Exercises.