Read This! The MAA Online book review column Understanding Analysis by Stephen D. Abbott Reviewed by Steve Kennedy

Teaching a one-semester introduction to real analysis can be a tricky business — on the one hand the entire point of the course is to construct, in excruciating detail, the theoretical underpinnings of calculus. On the other hand, going to such extraordinary lengths to prove theorems that the students already know to be true (and have never doubted) can seem to them a deliberate exercise in obfuscation. Let's face it, none of us has ever convinced a student that x² is continuous with an epsilon-delta argument. Of course, we don't really make the argument for that purpose; our real purpose (usually unstated) is to convince them that our definition of continuity is reasonable — the average student usually misses this point entirely. The obvious solution to the dilemma is to present the problems and the counter-intuitive examples that informed and motivated the theory in the first place. One can be tempted, and I admit I was tempted, to attempt a historically accurate presentation of the subject. There exists a terrific book by David Bressoud for those brave (or foolhardy) enough to take this radical approach. I think this is a mistake for a first pass through the theory and, if a student is only going to see this material once, she should get the cleaned-up, elegant, modern version. Future mathematicians, high-school teachers and the historically inclined can then be exposed to a historical treatment and be in a much better position to understand the evolution of the ideas if they are not also simultaneously trying to assimilate the technical details.

Thus a first course in analysis becomes a delicate balancing act between motivation and rigor. Steve Abbott's balance is nearly perfect. His text presents the standard topics of one-variable real analysis in the standard order. The distinguishing features are: the clear and easy prose style — this guy's writing is like a comfortable old shoe; the intuition-forward presentation — most concepts are tried out and walked around a little in the text before we get down to the nitty gritty; and the opening section of each chapter which presents a real problem that the rest of the chapter is designed to solve. Chapter two, infinite series, begins with a section on the pitfalls of rearrangement and asks what meaning can be attributed to a double summation. The chapter on continuity presents some (for us) old friends, Dirichlet's and Thomae's functions, and asks what sets can be sets of discontinuities of a function. Every chapter also concludes with i) a project section in which the chapter topic is explored more deeply but the proofs are only hinted at, filling in the missing details makes for a challenging exam or oral presentation project, and ii) a historical epilogue.

Of course no text is perfect and no mathematical book review is complete without quibbles: the standard proof of the chain rule is presented with no intuitive explanation, just a warning that it's a trick. The chapter on power series is motivated by a discussion of Francis Galton's use of branching processes. While interesting, this is somewhat ahistorical; I think Euler's computation of the sum of the reciprocals of the squares would be a much better choice (or, if that's too tired and overdone, maybe Newton's derivation of the binomial theorem and/or the arcsine series).

I should also mention that there are plenty of lovely exercises including many of a type I believe are particularly pedagogically potent, to wit: Construct an example of an object (function, sequence, series) that has property X (differentiability, convergence, absolute convergence), but does not have property Y (bounded derivative, reciprocals converge, absolute convergence when squared), or prove that no such object exists. (Oh, and a friendly tip — make sure you can do the exercise on the existence of uncountable anti-chains in the power set of N before you assign it, the author just might not respond to your panicked e-mail request for a hint before class meets!)

This terrific book will become the text of choice for the single-variable introductory analysis course; take a look at it next time you're preparing that class.

Publication Data: Understanding Analysis, by Stephen D. Abbott. Springer-Verlag, 2001. Hardcover, 257 pp., $39.95. ISBN 0-387-95060-5.

Also mentioned: A Radical Approach to Real Analysis, by David Bressoud. MAA, 1994. Paperbound, 336 pp., $36.50 ($29.00 to MAA members). ISBN 0-88385-701-4.

Steve Kennedy (skennedy@carleton.edu) is Associate Professor of Mathematics at Carleton College.

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