Counterexamples in Analysis [Paperback]

But it was a Faustian bargain, because immediately a host of bizarre and counterintuitive examples were discovered - continuous functions that were nowhere differentiable, nonmeasurable sets, one-to-one pairing of points between the line and the plane. These peculiar entities were deeply disturbing to many.

Poincare said "Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose... In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that."

These counterexamples displayed features that were nowhere to be found in the physical universe. When Richard Feynman was a physics graduate student at Princeton, he would tease his mathematician friends that mathematics was so easy that he could instantly decide the truth or falsehood of any mathematical statement they could give him. One day they challenged him with the grand-daddy of all the paradoxes, the Banach-Tarski paradox: That the unit ball in R3 could be divided into a finite number of pieces, and the pieces could, by rigid translation and rotation, be reassembled into two unit balls. But they made a mistake: instead of saying "unit ball in R3", they said "apple". Feynman quickly pointed out that the nonmeasurable pieces, that they had so rigorously defined, must split apart even every electron of the apple.

When I was a graduate student in mathematics, "Counterexamples in Analysis" was my favorite book, and I had a lot of fun amazing my fellow graduate students by quoting from it. Since then, however, I have swung around more to the viewpoint of Poincare and Feynman: "Logic sometimes makes monsters." From either viewpoint, however, the counterexamples are immensely entertaining.

It turns out that questions of the form "Does A always imply B?" entail proofs with two very different flavors, depending on whether the answer is affirmative or negative. The affirmative variety can be very difficult, as it usually deals with an infinity of things. But a negative answer requires only one solitary example of an A that is not a B; this is affectionately known as a "counter-example". These are the slickest little proofs around--often a one liner--and they can provide a lot of insight. Here's a trickier one: Are all linear functions continuous? Surprisingly, the answer is "no", which means there is a counter-example. Gelbaum and Olmsted show how to construct a discontinuous linear function. Case closed. They also provide examples of

A perfect nowhere dense set

A linear function space that is a lattice but not an algebra

A connected compact set that is not an arc

A divergent series whose general term approaches zero

A nonuniform limit of bounded functions that is not bounded

I won't give away any more (although there are hundreds). The book has chapters on real numbers, functions and limits, differentiation, sequences, infinite series, set and measure on the real axis, functions of two variables, metric and topological spaces, and more. Each section begins with a brief summary of the basic concepts and definitions, then launches into a list of terse counter-examples. This is simply indispensable for students of mathematical analysis, as it can help to explain why you cannot weaken those seemingly stringent hypotheses to various theorems; if you do, one of these quirky counter-examples will rush in and ruin your day. This is a great book to have on hand. I highly recommend it. (I won't tell you how it ends.)