由Trinity Colledge的William Trench编写，系统地叙述了实数系，一元微分，一元黎曼积分，实函数积分，度量空间。


目录：
Preface
Chapter 1 The Real Numbers 1
1.1 The Real Number System 1 1.2 Mathematical Induction 10 1.3 The Real Line 19


Chapter 2 Differential Calculus of Functions of One Variable 30
2.1 Functions and Limits 30 2.2 Continuity 53 2.3 Differentiable Functions of One Variable 73 2.4 L’Hospital’s Rule 88 2.5 Taylor’s Theorem 98

Chapter 3 Integral Calculus of Functions of One Variable 113
3.1 Definition of the Integral 113 3.2 Existence of the Integral 128 3.3 Properties of the Integral 135 3.4 Improper Integrals 151 3.5 A More Advanced Look at the Existence of the Proper Riemann Integral 171

Chapter 4 Infinite Sequences and Series 178
4.1 Sequences of Real Numbers 179 4.2 Earlier Topics Revisited With Sequences 195 4.3 Infinite Series of Constants 200
4.4 Sequences and Series of Functions 234 4.5 Power Series 257

Chapter 5 Real-Valued Functions of Several Variables 281

5.1 Structure of Rn 281 5.2 Continuous Real-Valued Function of n Variables 302 5.3 Partial Derivatives and the Differential 316 5.4 The Chain Rule and Taylor’s Theorem 339

Chapter 6 Vector-Valued Functions of Several Variables 361
6.1 Linear Transformations and Matrices 361 6.2 Continuity and Differentiability of Transformations 378 6.3 The Inverse Function Theorem 394 6.4. The Implicit Function Theorem 417


Chapter 7 Integrals of Functions of Several Variables 435
7.1 Definition and Existence of the Multiple Integral 435 7.2 Iterated Integrals and Multiple Integrals 462 7.3 Change of Variables in Multiple Integrals 484


Chapter 8 Metric Spaces 518
8.1 Introduction to Metric Spaces 518 8.2 Compact Sets in a Metric Space 535 8.3 Continuous Functions on Metric Spaces 543


Answers to Selected Exercises 549
Index 563