What is included with this book?
Preface | p. ix |
Introduction | p. 1 |
Basic Terminology | p. 1 |
Finite and Infinite Sets; Countably Infinite and Uncountably Infinite Sets | p. 7 |
Distance and Convergence | p. 10 |
Minicourse in Basic Logic | p. 15 |
Limit Points and Closure | p. 21 |
Review Problems for Chapter 1 | p. 23 |
Some Basic Topological Properties of R^{p} | p. 25 |
Unions and Intersections of Open and Closed Sets | p. 25 |
Compactness | p. 28 |
Some Applications of Compactness | p. 33 |
Least Upper Bounds and Completeness | p. 37 |
Review Problems for Chapter 2 | p. 42 |
Upper and Lower Limits of Sequences of Real Numbers | p. 45 |
Generalization of the Limit Concept | p. 45 |
Some Properties of Upper and Lower Limits | p. 49 |
Convergence of Power Series | p. 52 |
Review Problems for Chapter 3 | p. 56 |
Continuous Functions | p. 57 |
Continuity: Ideas, Basic Terminology, Properties | p. 57 |
Continuity and Compactness | p. 64 |
Types of Discontinuities | p. 70 |
The Cantor Set | p. 76 |
Review Problems for Chapter 4 | p. 80 |
Differentiation | p. 81 |
The Derivative and Its Basic Properties | p. 81 |
Additional Properties of the Derivative; Some Applications of the Mean Value Theorem | p. 86 |
Review Problems for Chapter 5 | p. 92 |
Riemann-Stieltjes Integration | p. 93 |
Definition of the Integral | p. 93 |
Properties of the Integral | p. 98 |
Functions of Bounded Variation | p. 106 |
Some Useful Integration Theorems | p. 111 |
Review Problems for Chapter 6 | p. 115 |
Uniform Convergence and Applications | p. 117 |
Pointwise and Uniform Convergence | p. 117 |
Uniform Convergence and Limit Operations | p. 122 |
The Weierstrass M-test and Applications | p. 125 |
Equicontinuity and the Arzela-Ascoli Theorem | p. 130 |
The Weierstrass Approximation Theorem | p. 134 |
Review Problems for Chapter 7 | p. 139 |
Further Topological Results | p. 141 |
The Extension Problem | p. 141 |
Baire Category Theorem | p. 144 |
Connectedness | p. 150 |
Semicontinuous Functions | p. 152 |
Review Problems for Chapter 8 | p. 158 |
Epilogue | p. 161 |
Some Compactness Results | p. 161 |
Replacing Cantor's Nested Set Property | p. 164 |
The Real Numbers Revisited | p. 165 |
Solutions to Problems | p. 167 |
Index | p. 207 |
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