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Product Code:  AMSTEXT/38 
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eBook ISBN:  9781470452896 
Product Code:  AMSTEXT/38.E 
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AMS Member Price:  $68.00 
Hardcover ISBN:  9781470451448 
eBook: ISBN:  9781470452896 
Product Code:  AMSTEXT/38.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 
Hardcover ISBN:  9781470451448 
Product Code:  AMSTEXT/38 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470452896 
Product Code:  AMSTEXT/38.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470451448 
eBook ISBN:  9781470452896 
Product Code:  AMSTEXT/38.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 38; 2019; 302 ppMSC: Primary 03; 97; 26
Real Analysis: A Constructive Approach Through Interval Arithmetic presents a careful treatment of calculus and its theoretical underpinnings from the constructivist point of view. This leads to an important and unique feature of this book: All existence proofs are direct, so showing that the numbers or functions in question exist means exactly that they can be explicitly calculated. For example, at the very beginning, the real numbers are shown to exist because they are constructed from the rationals using interval arithmetic. This approach, with its clear analogy to scientific measurement with tolerances, is taken throughout the book and makes the subject especially relevant and appealing to students with an interest in computing, applied mathematics, the sciences, and engineering.
The first part of the book contains all the usual material in a standard onesemester course in analysis of functions of a single real variable: continuity (uniform, not pointwise), derivatives, integrals, and convergence. The second part contains enough more technical material—including an introduction to complex variables and Fourier series—to fill out a fullyear course. Throughout the book the emphasis on rigorous and direct proofs is supported by an abundance of examples, exercises, and projects—many with hints—at the end of every section. The exposition is informal but exceptionally clear and well motivated throughout.
ReadershipUndergraduate and graduate students interested in real analysis, constructivism, and logic.

Table of Contents

Cover

Title page

Contents

Preface

Acknowledgments

Introduction

Chapter 0. Preliminaries

0.1. The Natural Numbers

0.2. The Rationals

Chapter 1. The Real Numbers and Completeness

1.0. Introduction

1.1. Interval Arithmetic

1.2. Families of Intersecting Intervals

1.3. Fine Families

1.4. Definition of the Reals

1.5. Real Number Arithmetic

1.6. Rational Approximations

1.7. Real Intervals and Completeness

1.8. Limits and Limiting Families

Appendix: The Goldbach Number and Trichotomy

Chapter 2. An Inverse Function Theorem and Its Applications

2.0. Introduction

2.1. Functions and Inverses

2.2. An Inverse Function Theorem

2.3. The Exponential Function

2.4. Natural Logs and the Euler Number e

Chapter 3. Limits, Sequences, and Series

3.1. Sequences and Convergence

3.2. Limits of Functions

3.3. Series of Numbers

Appendix I: Some Properties of Exp and Log

Appendix II: Rearrangements of Series

Chapter 4. Uniform Continuity

4.1. Definitions and Elementary Properties

4.2. Limits and Extensions

Appendix I: Are There NonContinuous Functions?

Appendix II: Continuity of DoubleSided Inverses

Appendix III: The Goldbach Function

Chapter 5. The Riemann Integral

5.1. Definition and Existence

5.2. Elementary Properties

5.3. Extensions and Improper Integrals

Chapter 6. Differentiation

6.1. Definitions and Basic Properties

6.2. The Arithmetic of Differentiability

6.3. Two Important Theorems

6.4. Derivative Tools

6.5. Integral Tools

Chapter 7. Sequences and Series of Functions

7.1. Sequences of Functions

7.2. Integrals and Derivatives of Sequences

7.3. Power Series

7.4. Taylor Series

7.5. The Periodic Functions

Appendix: Raabe’s Test and Binomial Series

Chapter 8. The Complex Numbers and Fourier Series

8.0. Introduction

8.1. The Complex Numbers ℂ

8.2. Complex Functions and Vectors

8.3. Fourier Series Theory

References

Index


Additional Material

Reviews

Here is another new textbook for undergraduate analysis, but it's far from a traditional approach. This one is dedicated to being entirely constructive. When we see constructivist approaches, we tend to think that they involve a lot of work to get basically the same thing, or even a bit less. This book might possibly convince you of their value.
Bill Satzer, MAA Reviews


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Real Analysis: A Constructive Approach Through Interval Arithmetic presents a careful treatment of calculus and its theoretical underpinnings from the constructivist point of view. This leads to an important and unique feature of this book: All existence proofs are direct, so showing that the numbers or functions in question exist means exactly that they can be explicitly calculated. For example, at the very beginning, the real numbers are shown to exist because they are constructed from the rationals using interval arithmetic. This approach, with its clear analogy to scientific measurement with tolerances, is taken throughout the book and makes the subject especially relevant and appealing to students with an interest in computing, applied mathematics, the sciences, and engineering.
The first part of the book contains all the usual material in a standard onesemester course in analysis of functions of a single real variable: continuity (uniform, not pointwise), derivatives, integrals, and convergence. The second part contains enough more technical material—including an introduction to complex variables and Fourier series—to fill out a fullyear course. Throughout the book the emphasis on rigorous and direct proofs is supported by an abundance of examples, exercises, and projects—many with hints—at the end of every section. The exposition is informal but exceptionally clear and well motivated throughout.
Undergraduate and graduate students interested in real analysis, constructivism, and logic.

Cover

Title page

Contents

Preface

Acknowledgments

Introduction

Chapter 0. Preliminaries

0.1. The Natural Numbers

0.2. The Rationals

Chapter 1. The Real Numbers and Completeness

1.0. Introduction

1.1. Interval Arithmetic

1.2. Families of Intersecting Intervals

1.3. Fine Families

1.4. Definition of the Reals

1.5. Real Number Arithmetic

1.6. Rational Approximations

1.7. Real Intervals and Completeness

1.8. Limits and Limiting Families

Appendix: The Goldbach Number and Trichotomy

Chapter 2. An Inverse Function Theorem and Its Applications

2.0. Introduction

2.1. Functions and Inverses

2.2. An Inverse Function Theorem

2.3. The Exponential Function

2.4. Natural Logs and the Euler Number e

Chapter 3. Limits, Sequences, and Series

3.1. Sequences and Convergence

3.2. Limits of Functions

3.3. Series of Numbers

Appendix I: Some Properties of Exp and Log

Appendix II: Rearrangements of Series

Chapter 4. Uniform Continuity

4.1. Definitions and Elementary Properties

4.2. Limits and Extensions

Appendix I: Are There NonContinuous Functions?

Appendix II: Continuity of DoubleSided Inverses

Appendix III: The Goldbach Function

Chapter 5. The Riemann Integral

5.1. Definition and Existence

5.2. Elementary Properties

5.3. Extensions and Improper Integrals

Chapter 6. Differentiation

6.1. Definitions and Basic Properties

6.2. The Arithmetic of Differentiability

6.3. Two Important Theorems

6.4. Derivative Tools

6.5. Integral Tools

Chapter 7. Sequences and Series of Functions

7.1. Sequences of Functions

7.2. Integrals and Derivatives of Sequences

7.3. Power Series

7.4. Taylor Series

7.5. The Periodic Functions

Appendix: Raabe’s Test and Binomial Series

Chapter 8. The Complex Numbers and Fourier Series

8.0. Introduction

8.1. The Complex Numbers ℂ

8.2. Complex Functions and Vectors

8.3. Fourier Series Theory

References

Index

Here is another new textbook for undergraduate analysis, but it's far from a traditional approach. This one is dedicated to being entirely constructive. When we see constructivist approaches, we tend to think that they involve a lot of work to get basically the same thing, or even a bit less. This book might possibly convince you of their value.
Bill Satzer, MAA Reviews