Examples and Problems in Advanced Calculus: Real-Valued Functions

Examples and Problems in Advanced Calculus: Real-Valued Functions delivers exactly what it promises. The book provides a plethora of advanced problems across (real-valued) calculus. It primarily focuses on material covered in undergraduate Calculus 1 and in Real Analysis, with an emphasis on the latter. The text is split into topics, with each topic including a subsection covering basic definitions and results; a set of challenging problems with worked solutions; and then exercises, loosely divided into “easier” and “harder” exercises to solve. As a professor, I look forward to using this book in order to find interesting and insightful problems. I would also suggest it to a particularly strong student of real analysis who wanted to push their boundaries on solving technical problems. However, I would be hesitant to give it a student seeing Real Analysis for the first time.

 

When teaching or studying topics that span calculus and real analysis, it can be a struggle to find problems of appropriate challenge and illumination. The author, citing 30 years of experience in teaching, tries to find an appropriate thread to follow for an advanced calculus class. The problems selected by the author are chosen to both challenge the reader and to illuminate subtle or intricate aspects of the material. The result is a solid reference for those who are comfortable with the subject matter but would prove challenging for those who are seeing or studying these subjects for the first time.

 

The text covers the usual range of topics in a one-semester course in real analysis: sets, limits and continuity, derivatives and applications, integrals and applications, and sequences and series. The text is organized into seven chapters along topical lines, and each chapter is further divided into four sections: Definitions, Problems, Exercises (easy), and Exercises (hard). Note that the “problems” have full solutions provided, while the exercises do not. Thus, one could imagine studying the problems to learn techniques of problem-solving, which could then be applied to the exercises. In practice, I suspect this process will be best utilized by those who are already familiar with the mathematical content this is meant to supplement (read: undergraduate real analysis). Having a mastery of the basic definitions and the simplest of exercises will allow readers to follow the more technical explanations present in the problem sets. For students who are struggling through Real Analysis for the first time, however, I worry these technical problems will prove too challenging. 

 

In summary: Davvaz’s text delivers exactly what it promises, offering problems in advanced calculus… with the keyword being “advanced” here. As a former and future teacher of undergraduate Real Analysis, I look forward to having this book on my desk to draw inspiration from. However, I would be hesitant to suggest it to any but my most advanced students.

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John Ross is an assistant professor of mathematics at Southwestern University.