Table of Contents for:

Understanding Analysis

Stephen Abbott, Middlebury College

Springer-Verlag Press, New York, 2001

Preface

1. The Real Numbers

1.1 Discussion: The Irrationality of sqrt(2)

1.2 Some Preliminaries

1.3 The Axiom of Completeness

1.4 Consequences of Completeness

1.5 Cantor's Theorem

1.6 Epilogue

2. Sequences and Series

2.1 Discussion: Rearrangements of Infinite Series

2.2 The Limit of a Sequence

2.3 The Algebraic and Order Limit Theorems

2.4 The Monotone Convergence Theorem and a First Look at Infinite Series

2.5 Subsequences and the Bolzano-Weierstrass Theorem

2.6 The Cauchy Criterion

2.7 Properties of Infinite Series

2.8 Double Summations and Products of Infinite Series

2.9 Epilogue

3 Basic Topology of R

3.1 Discussion: The Cantor Set

3.2 Open and Closed Sets

3.3 Compact Sets

3.4 Perfect Sets and Connected Sets

3.5 Baire's Theorem

3.6 Epilogue

4 Functional Limits and Continuity

4.1 Discussion: Examples of Dirichlet and Thomae

4.2 Functional Limits

4.3 Combinations of Continuous Functions

4.4 Continuous Functions on Compact Sets

4.5 The Intermediate Value Theorem

4.6 Sets of Discontinuity

4.7 Epilogue

5 The Derivative

5.1 Discussion: Are Derivatives Continuous?

5.2 Derivatives and the Intermediate Value Property

5.3 The Mean Value Theorem

5.4 A Continuous Nowhere-Differentiable Function

5.5 Epilogue

6 Sequences and Series of Functions

6.1 Discussion: Branching Processes

6.2 Uniform Convergence of Sequences of Functions

6.3 Uniform Convergence and Differentiation

6.4 Series of Functions

6.5 Power Series

6.6 Taylor Series

6.7 Epilogue

7 The Riemann Integral

7.1 Discussion: How should integration be defined?

7.2 The Definition of the Riemann Integral

7.3 Integrating Functions with Discontinuities

7.4 Properties of the Integral

7.5 The Fundamental Theorem of Calculus

7.6 Lebesgue's Criterion for Riemann Integrability

7.7 Epilogue

8 Additional Topics

8.1 The Generalized Riemann Integral

8.2 Metric Spaces and the Baire Category Theorem

8.3 Fourier Series

8.4 A Construction of R from Q