Understanding Analysis

Integral.- 7.3 Integrating Functions with Discontinuities.- 7.4 Properties of the Integral.- 7.5 The Fundamental Theorem of Calculus.- 7.6 Lebesgue's Criterion for Riemann Integrability.- 7.7 Epilogue.- 8 Additional Topics.- 8.1 The Generalized Riemann Integral.- 8.2 Metric Spaces and the Baire Category Theorem.- 8.3 Fourier Series.- 8.4 A Construction of R From Q.

without it." (L'enseignement mathématique, 48:3-4, 2002)"In writing Understanding Analysis, Stephen Abbott appreciates that modern readers may be unfamiliar with axiomatic arguments. Thus, many courses in analysis have been made easier by making them less interesting. The alternative is to make advanced topics accessible and focus on questions that give real analysis its value ... Prerequisite knowledge has been kept to a minimum: an understanding of single variable calculus. The text is written with introductory students in mind and will certainly be persuasive in establishing the value of analysis." (Nigel Steele, Times Higher Education Supplement, November 2002)"Each chapter begins with a discussion section and ends with an epilogue. The discussion serves to motivate the content of the chapter while the epilogue points tantalisingly to more advanced topics. ... I wish I had written this book! The development of the subject follows the tried-and-true path, but the presentation is engaging and challenging. Abbott focuses attention immediately on the topics which make analysis fascinating ... and makes them accessible to an inexperienced audience." (Scott Sciffer, The Australian Mathematical Society Gazette, 29:3, 2002)"In this book the author presents fundamental facts connected with the real functions of a real variable. ... In my opinion the didactic conception is interesting. The structure of the chapters is the following: First: discussion section (motivation, some questions or examples). Second: the fundamental text (also with some open questions) and a lot of exercises ... . Each chapter is closed by an Epilogue." (Ryszard Pawlak, Zentralblatt MATH, 966, 2001)"This book aims to build up the elements of analysis properly - that is, with a good level of understanding ... The author's intention was 'to restore intellectual liveliness of the course ...' In other words, he tries to present relatively advanced topics in an accessible way, and thus make the course interesting and worthy of effort ... This book will be useful for gifted students for supplementary reading to a basic analysis course or for students not specialising in mathematics who wish to put this material on a solid basis." (European Mathematical Society Newsletter, 42, December 2001)"The main goal of the author is to write an elementary one-semester book that exposes students to the rich rewards inherent in taking a mathematically rigorous approach to the study of functions of a real variable. This goal is excellently completed in this textbook. ... Each chapter begins with the discussion of some motivating examples and open questions. Each section contains many exercises ... . The book is warmly recommended to students, instructors interested in the study of introductory analysis." (József Németh, Acta Scientiarum Mathematicarum, 67, 2001)"This book is a mathematically rigorous approach to the study of functions of a real variable for student already familiar with the one variable calculus. ... focuses on seven topics based on a problem. ... Each of the topic begins thus with the discussion of a problem, continues with a clear mathematical development and ends with an epilogue on more advanced developments or considerations. ... The book is very clear, well presented, rigorous and should be recommended to undergraduate students as a good complement." (Yves Felix, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)