How to Think about Analysis

This book is an invaluable guide for any undergraduate student taking Analysis — the course usually called Real Analysis or Advanced Calculus. Designed as prerequisite reading material or to be used as a companion text, it provides strategies for learning pure mathematics, helps develop essential problem-solving skills, and seeks to strengthen student understanding of foundational topics in Analysis. Instructors will also find this book to be a useful resource when developing class notes, because it highlights common sources of student difficulty and confusion with the course content and provides methods with which to address them.

The author begins with a discussion of what Analysis is like to study, introduces mathematical notation, and explains how symbols and words should be used and read. Students who are new to studying pure mathematics will find this chapter especially helpful as they transition from computational-based classes to proof-based classes. The next two chapters relate axioms to definitions and theorems, outline how proofs are built from these components, and show how they are used to develop new theorems. Helpful and practical strategies for keeping up with the material, avoiding time-wasting, and getting help are presented in the proceeding chapter. The book then shifts focus to six core topics in Analysis; sequences, series, continuity, differentiability, integrability, and the real numbers. Each topic is defined and explained using carefully chosen examples combined with insightful diagrams. By the end of these chapters, the reader should be prepared to take on the challenging new problems assigned in their own Analysis class.

There are very few books on pure mathematics which I consider to be “page-turners,” but this book is definitely one of them. It is written using a friendly and informal tone yet carefully emphasizes and demonstrates the importance of paying attention to the details. It is an excellent read and is highly recommended for anyone interested in Analysis or any area of pure mathematics.

Stanley R. Huddy is currently an Assistant Professor of Mathematics in the Gildart Haase School of Computer Sciences and Engineering at Fairleigh Dickinson University. He earned his PhD at Clarkson University under the guidance of Joseph D. Skufca in 2013. His current interests are dynamic behaviors and synchronization patterns on networks of nonlinear systems as well as their applications, delay differential equations, inverse problems, combinatorics, and game theory.