The purpose of *A Guide to Real Variables* is to provide an aid and conceptual support for the student studying for the qualifying exam in real variables. Beginning with the foundations of the subject, the text moves rapidly but thoroughly through basic topics like completeness, convergence, sequences, series, compactness, topology and the like. All the basic examples like the Cantor set, the Weierstrass nowhere differentiable function, the Weierstrass approximation theory, the Baire category theorem, and the Ascoli-Arzela theorem are treated.

The book contains over 100 examples, and most of the basic proofs. It illustrates both the theory and the practice of this sophisticated subject. Graduate students studying for the qualifying exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful resource too.

Steven Krantz is well-known for his skill in expository writing and this volume confirms it. He is the author of more than 50 books, and more than 150 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize.

### Table of Contents

Preface

1. Basics

2. Sequences

3. Series

4. The Topology of the Real Line

5. Limits and the Continuity of Functions

6. The Derivative

7. The Integral

8. Sequences and Series of Functions

9. Advanced Topics

Glossary of Terms from Real Variable Theory

Bibliography

Index

About the Author

### About the Author

**Steven G. Krantz** was born in San Francisco, California and grew up in Redwood City, California. He received his undergraduate degree from the University of California at Santa Cruz and the PhD from Princeton University. Krantz has held faculty positions at UCLA, Princeton University, Penn State University, and Washington University in St. Louis. He is currently Deputy Director of the American Institute of Mathematics.

Krantz has written 160 scholarly papers and over 50 books. At least five of the latter are about aspects of complex analysis. Krantz is the holder of the Chauvenet Prize and the Beckenbach Book Award, both awarded by the Mathematical Association of America. He won the UCLA Alumni Association Distinguished Teaching Award. He is the author of *How to Teach Mathematics*. He has directed 16 PhD students. Krantz serves on the editorial boards of six journals and is Editor-in-Chief of two.

### MAA Review

This is the third book in the imaginatively devised series ‘MAA Guides’. These are not textbooks in the usual sense, and neither are they ‘handbooks’ consisting of lists of formulae, tables and definitions. They are intended for mathematics students in general and graduates and faculty in particular. Each book in the series provides an overview of a particular mathematical topic.

For example, what Steven Krantz has compiled here is a remarkably lucid and concise résumé of the main ideas and methods of real analysis. It begins with a review of set theory, functions, real numbers, and functions and it concludes with an outline of the main concepts and methods of metric space topology. Intermediate chapters cover a range of standard topics, such as sequences, series, topology of the real line, limits and continuity, the derivative and integral. Continued...

### All MAA Guides

1. A Guide to Complex Variables

2. A Guide to Advanced Real Analysis

3. A Guide to Real Variables

4. A Guide to Topology

5. A Guide to Elementary Number Theory

6. A Guide to Advanced Linear Algebra

7. A Guide to Plane Algebraic Curves

8. A Guide to Groups, Rings, and Fields

9. A Guide to Functional Analysis