Chapter 1 The Real and Complex Number Systems 1 Introduction 1 Ordered Sets 3 Fields 5 The Real Field 8 The Extended Real Number System 11 The Complex Field 12 Euclidean Spaces 16 Appendix 17 Exercises 21 Chapter 2 Basic Topology 24 Finite, Countable, and, Uncountable Sets 24 Metric Spaces 30 Compact Sets 36 Perfect Sets 41 Connected Sets 42 Exercises 43 Chapter 3 Numerical Sequences and Series 47 Convergent Sequences 47 Subsequences 51 Cauchy Sequences 52 Upper and Lower Limits 55 Some Special Sequences 57 Series 58 Series of Nonnegative Terms 61 The Number e 63 The Root and Ratio Tests 65 Power Series 69 Summation by Parts 70 Absolute Convergence 71 Addition and Multiplication of Series 72 Rearrangements 75 Exercises 78 Chapter 4 Continuity 83 Limits of Functions 83 Continuous Functions 85 Continuity and Compactness 89 Continuity and Connectedness 93 Discontinuities 94 Monotonic Functions 95 Infinite Limits and Limits at Infinity 97 Exercises 98 Chapter 5 Differetiation 103 The Derivative of a Real Function 103 Mean Value Theorems 107 The Continuity of Derivatives 108 L'Hospital's Rule 109 Derivatives of Higher Order 110 Taylor's Theorem 110 Differentiation of Vector-valued Functions 114 Chapter 6 The Riemann-Stieltjes Integral 120 Definition and Existence of the Integral 120 Properties of the Integral 128 Integration and Differentiation 133 Integration of Vector-valued Functions 135 Rectifiable Curves 136 Chapter 7 Sequences and Series of Functions 143 Discussion of Main Problem 143 Uniform Convergence 143 Uniform Convergence and Continuity 149 Uniform Convergence and Integration 151 Uniform Convergence and Differentiation 152 Equicontinuous Families of Functions 154 The Stone-Weierstrass Theorem 159 Exercises 165 Chapter 8 Some Special Functions 172 Power Series 172 The Exponential and Logarithmic Functions 178 The Trigonometric Functions 182 The Algebraic Completeness of the Complex Field 184 Fourier Series 185 The Gamma Function 192 Exericises 196 Chapter 9 Functions of Several Variables 204 Linear Transformations 204 Differentiation 211 The Contraction Principle 220 The Inverse Function Theorem 221 The Implicit Function Theorem 223 The Rank Theorem 228 Determinants 231 Derivatives of Higher Order 235 Differentiation of Integrals 236 Exercises 239 Chapter 10 Integration of Differential Forms 245 Integration 245 Primitive Mappings 248 Partitions of Unity 251 Change of Variables 252 Differential Forms 253 Simplexes and Chains 266 Stokes' Theorem 273 Closed Forms and Exact Forms 275 Vector Analysis 280 Exercises 288 Chapter 11 The Lebesgue Theory 300 Set Functions 300 Construction of the lebesgue Measure 302 Measure Spaces 310 Measurable Functions 310 Simple Functions 313 Integration 314 Comparison with the Riemann Integral 322 Integration of Complex Functions 325 Functions of Class L2 325 Exercises 332 Bibliography 335 List of Special Symbols 337 Index 339