| Preface |
| 1 Real Numbers |
| 1.1. Set-Theoretic Preliminaries |
| 1.2. Axioms for the Real Number System |
| 1.3. Consequences of the Addition Axioms |
| 1.4. Consequences of the Multiplication Axioms |
| 1.5. Consequences of the Order Axioms |
| 1.6. Consequences of the Least Upper Bound Axiom |
| 1.7. The Principle of Archimedes and Its Consequences |
| 1.8. The Principle of Nested Intervals |
| 1.9. The Extended Real Number System |
| Problems |
| 2 Sets |
| 2.1. Operations on Sets |
| 2.2. Equivalence of Sets |
| 2.3. Countable Sets |
| 2.4 Uncountable Sets |
| 2.5. Mathematical Structures |
| 2.6. n-Dimensional Space |
| 2.7. Complex Numbers |
| 2.8. Functions and Graphs |
| Problems |
| 3 Metric Spaces |
| 3.1. Definitions and Examples |
| 3.2. Open Sets |
| 3.3. Convergent Sequences and Homeomorphisms |
| 3.4. Limit Points |
| 3.5. Closed Sets |
| 3.6. Dense Sets and Closures |
| 3.7. Complete Metric Spaces |
| 3.8. Completion of a Metric Space |
| 3.9. Compactness |
| Problems |
| 4 Limits |
| 4.1. Basic Concepts |
| 4.2. Some General Theorems |
| 4.3. Limits of Numerical Functions |
| 4.4. Upper and Lower Limits |
| 4.5. Nondecreasing and Nonincreasing Functions |
| 4.6. Limits of Numerical Functions |
| 4.7. Limits of Vector Functions |
| Problems |
| 5 Continuous Functions |
| 5.1. Continuous Functions on a Metric Space |
| 5.2. Continuous Numerical Functions on the Real Line |
| 5.3. Monotonic Functions |
| 5.4. The Logarithm |
| 5.5. The Exponential |
| 5.6. Trignometric Functions |
| 5.7. Applications of Trigonometric Functions |
| 5.8. Continuous Vector Functions of a Vecor Variable |
| 5.9. Sequences of Functions |
| Problems |
| 6 Series |
| 6.1. Numerical Series |
| 6.2. Absolute and Conditional Convergences |
| 6.3. Operations on Series |
| 6.4. Series of Vectors |
| 6.5. Series of Functions |
| 6.6. Power Series |
| Problems |
| 7 The Derivative |
| 7.1. Definitions and Examples |
| 7.2. Properties of Differentiable Functions |
| 7.3. The Differential |
| 7.4. Mean Value Theorems |
| 7.5. Concavity and Inflection Points |
| 7.6. L'Hospital's Rules |
| Problems |
| 8 Higher Derivatives |
| 8.1. Definitions and Examples |
| 8.2. Taylor's Formula |
| 8.3. More on Concavity and Inflection P |
| 8.4. Another Version of Taylor's Formula |
| 8.5. Taylor Series |
| 8.6. Complex Exponentials and Trigonometric Functions |
| 8.7. Hyperbolic Functions |
| Problems |
| 9 The Integral |
| 9.1. Definitions and Basic Properties |
| 9.2. Area and Arc Length |
| 9.3. Antiderivatives and Indefinite Integrals |
| 9.4. Technique of Indefinite Integrals |
| 9.5. Evaluation of Definite Integrals |
| 9.6. More on Area |
| 9.7. More on Arc Length |
| 9.8. Area of a Surface of Revolution |
| 9.9. Further Applications of Integration |
| 9.10. Integration of Sequences of Functions |
| 9.11. Parameter-Dependent Integrals |
| 9.12. Line Integrals |
| Problems |
| 10 Analytic Functions |
| 10.1. Basic Concepts |
| 10.2. Line Integrals of Complex Functions |
| 10.3. Cauchy's Theorem and Its Consequences |
| 10.4. Residues and Isolated Singular Points |
| 10.5. Mappings and Elementary Functions |
| Problems |
| 11 Improper Integrals |
| 11.1. Improper Integralsof the First Kind |
| 11.2. Convergence of Improper Integrals |
| 11.3. Improper Integrals of the Second and Third Kinds |
| 11.4 Evaluation of Improper Integrals by Residues |
| 11.5 Parameter-Dependent ImproperIntegrals |
| 11.6 The Gamma and Beta Functions |
| Problems |
| Appendix A Elementary Symbolic Logic |
| Appendix B Measure and Integration on a Compact Metric Space |
| Selected Hints and Answers |
| Index |