The fourth edition of this classic introduction to analysis retains the freshness of the first edition as well as its charming conversational style. This augmented and updated edition was completed by Harold Boas after his father’s death. He calls the book “an heirloom and a memorial.”
As originally published this book had two parts — chapters called “Sets” and “Functions”. The latter took the discussion up through the analysis of continuous and differentiable functions, but stopped short of any treatment of integration. In his preface to the third edition, the author notes that he left out integration “reluctantly, because of the many technical details that are needed before one gets to the interesting results.” He did, however, produce a draft of a chapter on integration (without all the technical details) on the grounds that “one need not understand the inner workings of the motor to appreciate a drive in the country.” Indeed, it turns out to be a very pleasant ride. Harold Boas reworked his father’s draft and added notes, exercises and solutions for what became the third chapter in this fourth edition.
This is not in any way a traditional textbook. It is more like a series of informal lectures, wordy, chatty and not the least bit concise. The author’s aim, and the book’s great strength, is to bring back a sense of wonder to a subject that, in his opinion, had been lost. The intended reader should have had a course in calculus. Nothing more — other than perhaps a small dose of that elusive thing called mathematical maturity — is called for. Typically the author starts slowly but the level of difficulty rises steeply. The author advises the reader to skip forward if the going is too tough, and he makes this approach workable.
Here are some elements of the book that I found especially notable: the treatment of the Baire Category theorem and its use to prove the existence of a continuous, ever-oscillating function; singular functions and an example of two functions with the same derivative that do not differ by a constant; the universal chord theorem for periodic continuous functions; and the Riesz representation theorem for Stieltjes integrals.
The level of proof varies considerably throughout: sometimes detailed proofs, sometimes sketches, sometimes “it can be shown”. Exercises abound; they range from items essential to the text to the illustrative. Solutions to all the exercises are provided. The notes at the end of many sections are a true highlight. They are full of stories, references, and connections.
The intended audience for this book is the neophyte. As such, the book could be used for independent study or as a supplement to a more standard analysis text. Much of the material would also be of interest to more advanced students. Indeed, this is a book for anyone interested in renewing their sense of wonder in analysis.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.