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| Preface | |
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| Acknowledgments | |
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| Elementary Calculus | |
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| Preliminary Concepts | |
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| Limits and Continuity | |
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| Differentiation | |
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| Integration | |
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| Sequences and Series of Constants | |
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| Power Series and Taylor Series | |
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| Summary | |
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| Exercises | |
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| Interlude: Fermat, Descartes, and the Tangent Problem | |
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| Introduction to Real Analysis | |
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| Basic Topology of the Real Numbers | |
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| Limits and Continuity | |
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| Differentiation | |
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| Riemann and Riemann-Stieltjes Integration | |
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| Sequences, Series, and Convergence Tests | |
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| Pointwise and Uniform Convergence | |
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| Summary | |
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| Exercises | |
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| Interlude: Euler and the "Basel Problem" | |
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| A Brief Introduction to Lebesgue Theory | |
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| Lebesgue Measure and Measurable Sets | |
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| The Lebesgue Integral | |
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| Measure, Integral, and Convergence | |
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| Littlewood's Three Principles | |
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| Summary | |
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| Exercises | |
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| Interlude: The Set of Rational Numbers Is Very Large and Very Small | |
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| Special Topics | |
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| Modeling with Logistic Functions-Numerical Derivatives | |
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| Numerical Quadrature | |
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| Fourier Series | |
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| Special Functions-The Gamma Function | |
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| Calculus Without Limits: Differential Algebra | |
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| Summary | |
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| Exercises | |
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| Definitions & Theorems of Elementary Real Analysis | |
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| Limits | |
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| Continuity | |
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| The Derivative | |
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| Riemann Integration | |
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| Riemann-Stieltjes Integration | |
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| Sequences and Series of Constants | |
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| Sequences and Series of Functions | |
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| A Brief Calculus Chronology | |
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| Projects in Real Analysis | |
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| Historical Writing Projects | |
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| Induction Proofs: Summations, Inequalities, and Divisibility | |
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| Series Rearrangements | |
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| Newton and the Binomial Theorem | |
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| Symmetric Sums of Logarithms | |
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| Logical Equivalence: Completeness of the Real Numbers | |
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| Vitali's Nonmeasurable Set | |
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| Sources for Real Analysis Projects | |
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| Sources for Projects for Calculus Students | |
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| Bibliography | |
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| Index | |