What’s the best part of mathematics? One answer might be: counterexamples. As the authors of this book say (p. vii), “It has been our experience that a mathematical question resolved by a counterexample has the pungency of good drama”

What’s not to like? I have some favorites, many of which need plowing through, such as (p. 38) “a function that is everywhere continuous and nowhere differentiable” and (p. 30) “a function whose points of discontinuity form an arbitrary given F_{σ} set” (meaning a countable union of closed sets). Then there’s the educational (perhaps too educational to some first-year calculus students) example (p. 36) of a “differentiable function having an extreme value at a point where the derivative does not make a simple change of sign… In no interval of the form (a,0) or (0,b) is f monotonic” and (p. 59) a “series for which the ratio test fails”, followed (p. 60) by a “series for which the root test fails”; I enjoyed the interesting combining of geometric series. And we mustn’t omit the “startling result… due to W. Sierpinski” (pp. 74–75) that there exists a single power series which, with proper placement of parentheses, converges uniformly to any given function continuous on [0,1].

The series ∑ n!x^{n} provides an example of a power series convergent at only one point (x = 0). It is familiar to many. But I, for one, didn’t know that this is the Maclaurin series for an actual closed-form function. For details see pp. 69–70 of this book.

I’ll mention only a few geometric gems. On p. 138, “three disjoint plane regions with a common frontier”. And of course there are Cantor sets of various measures (pp. 85, 88) and Hilbert’s space-filling curve (p. 133). But how about (p. 135) “a simple [not crossing itself] arc in the unit square and of plane measure arbitrarily near 1” and (p. 149) “a simple closed curve whose plane measure is greater than that of the bounded region that it encloses”. And p. 150 is of interest to any calculus student who truly wants to begin to understand the concept of surface area. P. 161 shows that sequential convergence does not define a topology, nor (p. 164) are all limit points in topological spaces the limits of sequences. (But the example makes no mention of nets.)

The authors’ wisely chose to organize the book into chapters titled “The Real Number System”, “Functions and Limits”, “Differentiation”, “Riemann Integration”, “Sequences”, “Infinite Series”, “Uniform Convergence”, “Sets and Measure on the Real Axis”, “Functions of Two Variables”, “Plane Sets”, “Area”, and “Metric and Topological Spaces”. This gives us a good sense, not only of the entirely of the counterexamples involved, but a huge chunk of math itself.

I learned a few things. I had not known about Toeplitz matrices, and I would not have guessed that they had to do with sequences. This was but one instance of a counterexample to something I hadn’t known needed a counterexample to!

This is not an easy book, especially for those who truly want to *see* the counterexamples. It’s written mostly in the style of journal articles; we have to provide the details. (p. 135 was the hardest going for me, even with three lovely diagrams provided.) Especially impressive were those instances of multiple counterexamples, accompanied by phrases such as “A third construction of a Cantor set… (p. 89), “A more extreme case…” (p. 80), and “A second method of constructing a simple arc with positive plane measure…” Moreover, the authors continue, “this is somewhat simpler conceptually than the construction just described, but has the disadvantage that certain subintervals of [0, 1] are mapped onto sets of zero plane measure…” It seems that just about any possible refinement or generalization or related theorem is in there. This book is certainly quite thorough.

There are many additions to the counterexamples — refinements and answers to further questions — where no author was mentioned. I wondered whether it was the authors who deserve the credit for these, and if so that’s just another kudo to them. Only once did they leave me hanging: On p. 176 they provide an example of three functions, all of which are semicontinuous everywhere (upper or lower) but the sum is nowhere semicontinuous. I wondered whether there are *two* such functions.

I have only a handful of nitpicks, and I could be mistaken about every one of them. On pp. 89–90 (“a perfect nowhere dense set of irrational numbers”) the authors begin their construction: “… by making use of a sequence {r_{n}} whose terms constitute the set of all rational numbers of (0,1). Start as in the definition of the Cantor set C, but extend the open interval so that the center remains at ½, so that its endpoints are irrational and so that the enlarged open interval contains the point r_{1}.” I thought, in anticipation, “But it would also have to contain other rationals, perhaps r_{2} …” They continue, “At the second stage remove from each of the two remaining closed intervals an enlarged open middle ‘third’ in such a way that the midpoints remain midpoints, the endpoints are irrational, and the second rational number r_{2} is removed.” “Huh?” I thought. “Suppose r_{2} was already removed in the first stage?” It seems to me that we need here some refinement of their example; perhaps they mean to remove the first rational number that was not already removed?

On pp. 170–171 I wondered why “a decomposition of a three-dimensional Euclidean ball B into five disjoint subsets S_{1}, S_{2}, S_{3}, S_{4}, S_{5}… and five rigid motions, R_{1}, R_{2}, R_{3}, R_{4}, R_{5} such that B ≅ R_{1}(S_{1}) ≅ R_{2}(S_{2}) ∪ R_{3}(S_{3}) ∪ R_{4}(S_{4}) ∪ R_{5}(S_{5})” did not lead to mention of the Banach-Tarski “paradox”.

The following nitpicks are definitely on me: I was disappointed when proofs (or even the examples themselves) were omitted, in favor of references. (E.g., pp. 73 and 153). However, I understand the necessity for this; some pathological functions are downright psychotic! Also, sometimes, when they referred readers to other examples in the book, I felt just-plain lazy! But not only would repetition be a space-taker-upper, but also referencing emphasizes the connections within math.

The book provides a very enjoyable review of many undergraduate and graduate level courses. How many books cover so much material in 179 pages? Could, in fact, this book be used not only as supplementary reading for many courses, but also as an actual text? I can envision innovative courses built around it (or parts of it), mathematically astute and enthusiastic students using the various counterexamples as a basepoint, the instructor explaining (or the students determining) what they’re counterexamples *to, *why they’re so surprising (why they’re considered counterexamples*). *

As I write this, it occurs to me that many items in this book are not specifically presented as counterexamples as such. For example, the above-mentioned “function whose points of discontinuity form an arbitrary F_{σ} set”. What conjecture does that negate? What cherished notion does it squelch? What equally cherished territory arises in its place? To me these questions provided a pleasant and illuminating challenge and perhaps a deeper meaning to the book.

Marion Cohen teaches math at Arcadia University in Glenside, PA. A course she developed, “Truth and Beauty: Mathematics in Literature” has been increasingly popular with students. An article about it appears in the Dec 2010/Jan 2011 issue of MAA FOCUS. She is the author of Crossing the Equal Sign, a poetry book about the experience of math.