Preface
Part I: Functions of One Variable
1. The Real Number System
1.1 Algebraic Properties of R
1.2 Order Properties of R
1.3 Completeness Property of R
1.4 Mathematical Induction
1.5 Euclidean Space
2. Numerical Sequences
2.1 Limits of Sequences
2.2 Monotone Sequences
2.3 Subsequences and Cauchy Sequences
2.4 Limits Inferior and Superior
3. Limits and Continuity on R
3.1 Limit of a Function
3.2 Limits Inferior and Superior
3.3 Continuous Functions
3.4 Properties of Continuous Functions
3.5 Uniform Continuity
4. Differentiation on R
4.1 Definition of Derivative and Examples
4.2 The Mean Value Theorem
4.3 Convex Functions
4.4 Inverse Functions
4.5 L’Hospital’s Rule
4.6 Taylor’s Theorem on R
4.7 Newton’s Method
5. Riemann Integration on R
5.1 The Riemann–Darboux Integral
5.2 Properties of the Integral
5.3 Evaluation of the Integral
5.4 Stirling’s Formula
5.5 Integral Mean Value Theorems
5.6 Estimation of the Integral
5.7 Improper Integrals
5.8 A Deeper Look at Riemann Integrability
5.9 Functions of Bounded Variation
5.10 The Riemann–Stieltjes Integral
6. Numerical Infinite Series
6.1 Definition and Examples
6.2 Series with Nonnegative Terms
6.3 More Refined Convergence Tests
6.4 Absolute and Conditional Convergence
6.5 Double Sequences and Series
7. Sequences and Series of Functions
7.1 Convergence of Sequences of Functions
7.2 Properties of the Limit Function
7.3 Convergence of Series of Functions
7.4 Power Series
Part II: Functions of Several Variables
8. Metric Spaces
8.1 Definitions and Examples
8.2 Open and Closed Sets
8.3 Closure, Interior, and Boundary
8.4 Limits and Continuity
8.5 Compact Sets
8.6 The Arzelà–Ascoli Theorem
8.7 Connected Sets
8.8 The Stone–Weierstrass Theorem
8.9 Baire’s Theorem
9. Differentiation on Rn
9.1 Definition of the Derivative
9.2 Properties of the Differential
9.3 Further Properties of the Differential
9.4 Inverse Function Theorem
9.5 Implicit Function Theorem
9.6 Higher Order Partial Derivatives
9.7 Higher Order Differentials and Taylor’s Theorem
9.8 Optimization
10. Lebesgue Measure on Rn
10.1 General Measure Theory
10.2 Lebesgue Outer Measure
10.3 Lebesgue Measure
10.4 Borel Sets
10.5 Measurable Functions
11. Lebesgue Integration on Rn
11.1 Riemann Integration on Rn
11.2 The Lebesgue Integral
11.3 Convergence Theorems
11.4 Connections with Riemann Integration
11.5 Iterated Integrals
11.6 Change of Variables
12. Curves and Surfaces in Rn
12.1 Parameterized Curves
12.2 Integration on Curves
12.3 Parameterized Surfaces
12.4 m-Dimensional Surfaces
13. Integration on Surfaces
13.1 Differential Forms
13.2 Integrals on Parameterized Surfaces
13.3 Partitions of Unity
13.4 Integration on Compact m-Surfaces
13.5 The Fundamental Theorems of Calculus
13.6 Closed Forms in Rn
Part III: Appendices
A Set Theory
B Linear Algebra
C Solutions to Selected Problems
Biography
Hugo D. Junghenn is emeritus professor of mathematics at The George Washington University. He has published numerous journal articles and is the author of several books, including Option Valuation: A First Course in Financial Mathematics; Principles of Analysis; and Discrete Mathematics with Coding. His research interests include functional analysis, semigroups, and probability.
