Elementary Real Analysis by Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner is a comprehensive guide to the fundamentals of real analysis, a branch of mathematics that deals with the properties of real numbers and their sequences and series. The book provides an introduction to the basic concepts and tools of analysis, such as limits, continuity, differentiation, integration, and series, with an emphasis on clarity, rigour, and accessibility.

The authors begin with a review of the axioms and properties of the real number system and the basic topology of the real line, including the concepts of open and closed sets, convergence, and compactness. They then move on to the study of functions, starting with the notion of limits and continuity, and proceeding to the derivative and the integral. The authors provide numerous examples, exercises, and problems to reinforce the concepts and techniques introduced in each chapter.

The book also covers more advanced topics such as uniform convergence, sequences and series of functions, and the Riemann-Stieltjes integral, which extends the classical Riemann integral to include functions that are not necessarily continuous. The authors present these topics in a clear and concise manner, with numerous examples and exercises to help the reader gain a deeper understanding of the subject.

One of the strengths of Elementary Real Analysis is the authors’ focus on applications of analysis to other areas of mathematics, including topology, differential equations, and Fourier series. They show how the techniques and concepts of real analysis can be used to solve problems in these areas and provide numerous examples and exercises to illustrate the applications.

The book is written in a clear and engaging style, with many helpful diagrams and illustrations to aid in the understanding of the material. The authors also provide historical notes and biographies of mathematicians who have made important contributions to the development of real analysis, giving the reader a sense of the intellectual history of the subject.

Overall, Elementary Real Analysis is an excellent introduction to the fundamentals of real analysis, suitable for undergraduate students in mathematics, physics, and engineering, as well as for self-study by anyone interested in the subject. The book’s clear and accessible style, numerous examples and exercises, and focus on applications make it an invaluable resource for anyone seeking to master the techniques and concepts of real analysis.