Read This! The MAA Online book review column Problems in Mathematical Analysis II Continuity and Differentiation by W.J. Kaczor and M.T. Nowak Reviewed by Morteza Seddighin

Although one cannot find any statement, which is labeled as a "Theorem" in the book, there are scores of theorems, which are tagged as problems. In each section of the book, a segment consisting of pertinent definitions precedes the problem statements. The authors give the simplest definition for each mathematical concept. The style of presentation of the content in this book is proven to be a motivating approach in constructing and conveying mathematical knowledge. The authors give the reader an opportunity to first solve each individual problem by himself and then contrast his solution with the one provided in the book. This often leads the readers to find new solutions to the problems and hence boosts their ability to carry out further research in mathematical analysis. The authors discuss the problems and their solutions in a manner that incites the reader to explore ways to generalize them.

The book contains many different kinds of problems. Some ask for examples and counter examples while others ask the reader to classify functions possessing a certain property. Some are computational and application problems. There are numerous inequalities (some famous and some not so famous) stated in the book whose proofs depend on properties of convex functions or the traits of differentiation. Although the solutions are provided in the second part of the book, the authors have left enough gaps in them to challenge the reader who is not able to come up with a complete solution of his own. There are numerous occasions that the authors use phrases such as "One can show that..." or "...is verifiable." Each section of the book starts with a number of problems that are moderate in difficulty but the level of difficulty gradually grows until we reach some very challenging problems.

For the most part, the book looks like a traditional book on mathematical analysis in that it treats real functions of one real variable. However, it also covers some important definitions and concepts of modern analysis on metric spaces. This includes the open mapping theorem, completeness, compactness, and the uniform boundedness theorem. There are also occasions that the solutions presented in the book require some knowledge of functional analysis. A good example of this is when the authors present an elegant proof for the famous Faa di Bruno formula. This proof, which is given by S. Roman, first appeared in The American Mathematical Monthly in 1980. Here, Roman considers the set of linear functionals defined on the space of all polynomials with real coefficients.

What is notable about this book is its through coverage of some topics that are covered very briefly in other compatible books. This includes topics such as functional equations and convex functions.

This book will be of interest to anyone who wishes to pursue research in mathematical analysis and its applications. It is also excellent for students who want to enhance their skills in real analysis to support their research in other fields such as functional analysis and complex analysis. This book would be a useful supplement to any graduate textbook in mathematical analysis, and some problems are also suitable for undergraduate students.

See also Problems in Mathematical Analysis I, discussed in the December 2000 Briefly Noted column.

Publication Data: Problems in Mathematical Analysis II: Continuity and Differentiation, by W.J. Kaczor and M.T. Nowak. American Mathematical Society, 2001. Paperback, 395 pp, $49.00 ($39.00 to AMS members). ISBN 0-8218-2051-6

Morteza Seddighin (mseddigh@indiana.edu) is an associate professor of mathematics at Indiana University East. His research interests are functional analysis and operator theory.

Go to... The main Read This! page The index of all reviews The MAA Online front page

Find out... About the Mathematical Association of America Why you should be a member of the MAA How you can help support the activities of the MAA

Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouvêa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.