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Counterexamples in Calculus

Mathematical Association of America
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This is a valuable but thinly-described compendium of counterexamples in single-variable calculus. The book is vague about who it is aimed at, sometimes addressing the student and sometimes the teacher. I believe it is most useful as a source of enrichment activities and would be read by the teacher and not by the student.

The present volume is a reprint of a work published in Australia in 2004. It claims to be modeled on Gelbaum and Olmsted’s Counterexamples in Analysis, but the two books have very different natures. Gelbaum and Olmsted modestly state that their book is “meant primarily for browsing” (p. vi) and “we hope that the readers of this book will find both enjoyment and stimulation from this collection” (p. vii). Klymchuk’s book claims much stronger benefits, stating that “using counterexamples fosters discovery and makes learning more active” and that “students reported that it helped them to understand concepts better, prevent mistakes, develop logical and critical thinking, and made learning mathematics more challenging, interesting and creative.” (p. 5)

Gelbaum and Olmsted’s book is discursive, with an introduction to each section giving definitions and context, and with references to other results supplementing most counterexamples. Klymchuk’s book is very austere, and this is its big weakness: It consists of a request to show through counterexamples the falsity of a long list of statements, followed by an counterexample for each statement. There is no indication of how one might fix up the statements to make them true, how topics can be related related to other topics, or how one might discover these counterexamples by oneself. It’s hard to imagine any unaided calculus student getting more than a few statements into this before giving up. It’s more suited for teachers, but unless the teacher is very knowledgeable about calculus, he might not get very far either.

Some of the counterexamples depend on a careful reading of the definition, but these are problematic because the book does not define anything. For example, item 1.3 asks for a counterexample to “A quadratic function of x is one in which the highest power of x is two.” The counterexamples given are x2 + √x and x2 + x – 1/x. I thought this was a trick question, since it requires us change meanings halfway through, by interpreting “quadratic function” as “quadratic polynomial” and “one in which the highest power of x is two” as “function that need not be a polynomial”.

The best use of counterexamples I have seen in calculus is Edmund Landau’s Differential and Integral Calculus. Normally when Landau states a theorem, he also shows by counterexamples that no part of the hypothesis can be omitted without falsifying the result. Most of the counterexamples in Klymchuk’s book seem to have been generated in the same way, but we are not told what theorem they are testing. Context makes all the difference.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

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Monday, January 25, 2010
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Sergiy Klymchuk
Classroom Resource Materials
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Allen Stenger
Counterexamples in Calculus

1. Functions
2. Limits
3. Continuity
4. Differential Calculs
5. Integral Calculus

Suggested Solutions
1. Functions
2. Limits
3. Continuity
4. Differential Calculs
5. Integral Calculus

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Saturday, April 3, 2010