**AMS/MAA Textbooks**

Volume: 70;
2022;
360 pp;
Softcover

MSC: Primary 26; 37; 39;

**Print ISBN: 978-1-4704-6454-7
Product Code: TEXT/70**

List Price: $69.00

AMS Member Price: $51.75

MAA Member Price: $51.75

**Electronic ISBN: 978-1-4704-6847-7
Product Code: TEXT/70.E**

List Price: $69.00

AMS Member Price: $51.75

MAA Member Price: $51.75

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#### Supplemental Materials

# Welcome to Real Analysis: Continuity and Calculus, Distance and Dynamics

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*Benjamin B. Kennedy*

MAA Press: An Imprint of the American Mathematical Society

Welcome to Real Analysis is designed for use in an
introductory undergraduate course in real analysis. Much of the
development is in the setting of the general metric space. The book
makes substantial use not only of the real line and
\(n\)-dimensional Euclidean space, but also sequence and
function spaces. Proving and extending results from single-variable
calculus provides motivation throughout. The more abstract ideas come
to life in meaningful and accessible applications. For example, the
contraction mapping principle is used to prove an existence and
uniqueness theorem for solutions of ordinary differential equations
and the existence of certain fractals; the continuity of the
integration operator on the space of continuous functions on a compact
interval paves the way for some results about power series.

The exposition is exceedingly clear and well-motivated. There are
a wide variety of exercises and many pedagogical innovations. For
example, each chapter includes Reading Questions so that students can
check their understanding. In addition to the standard material in a
first real analysis course, the book contains two concluding chapters
on dynamical systems and fractals as an illustration of the power of
the theory developed.

#### Readership

Undergraduate students interested in learning real analysis.

#### Table of Contents

# Table of Contents

## Welcome to Real Analysis: Continuity and Calculus, Distance and Dynamics

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents vii9
- Preface xi13
- Chapter 0. Where We’re Starting and Where We’re Going 115
- Chapter 1. Essential Tools 519
- Chapter 2. Metric Spaces 4155
- 2.1. The definition of a metric space 4155
- 2.2. Important metrics in Rⁿ 4559
- 2.3. Open balls and open sets in metric spaces 4862
- 2.4. Closed sets and limit points 5367
- 2.5. Interior, closure, and boundary 5973
- 2.6. Dense subsets 6276
- 2.7. Equivalent metrics 6377
- 2.8. Normed vector spaces 6882
- 2.9. A brief note about conventions 7185
- 2.10. Exercises 7185

- Chapter 3. Sequences 7791
- 3.1. Convergence 7892
- 3.2. Discrete dynamical systems 87101
- 3.3. Sequences and limit points 92106
- 3.4. Algebraic theorems for sequences 95109
- 3.5. Subsequences 101115
- 3.6. Completeness 107121
- 3.7. The contraction mapping principle 110124
- 3.8. Sets of sequences as metric spaces 119133
- 3.9. Exercises 125139

- Chapter 4. Continuity 131145
- 4.1. The definition of continuity 131145
- 4.2. Equivalent formulations of continuity 142156
- 4.3. Continuity and limit theorems for scalar- valued functions 146160
- 4.4. Continuity and products of metric spaces 150164
- 4.5. Uniform continuity 154168
- 4.6. The metric space 𝐶([𝑎,𝑏],R ) 160174
- 4.7. An application to functional equations 167181
- 4.8. Exercises 170184

- Chapter 5. Compactness and Connectedness 177191
- 5.1. Basic definitions and results on compactness 177191
- 5.2. The nested set property for compact sets 181195
- 5.3. Compactness and continuity 183197
- 5.4. Other facts about compactness 185199
- 5.5. Connectedness 191205
- 5.6. Periodic points of maps on intervals 195209
- 5.7. Injective continuous functions defined on intervals 200214
- 5.8. Exercises 202216

- Chapter 6. The Derivative 209223
- Chapter 7. The Riemann Integral 235249
- Chapter 8. Sequences of Functions 261275
- Chapter 9. Chaos in Discrete Dynamical Systems 301315
- Chapter 10. The Hausdorff Metric and Fractals 341355
- Bibliography 355369
- Index 357371
- Back Cover Back Cover1375