On a discontinuity of a derivative

Authors

John Klippert

Type: Article

Publication Date: 2000-03-01

Citations: 3

DOI: https://doi.org/10.1080/00207390050032252

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Abstract

Many texts in 'advanced calculus' present Darboux's Theorem (also known as the Intermediate Value Theorem for Derivatives) and the well-known example f(x) = { x2sin1/x, x = 0 /0, x = 0 of a function with discontinuous derivative at the origin. But these texts typically fail to discuss the relationship between Darboux's result and the type of discontinuity a given derivative must have at such a point. It is no accident, for example, that the discontinuity of f'(x) = { 2xsin 1/x - cos 1/x, x = 0 /0, x = 0 at the origin is such that limx-0f'(x) does not exist. In this paper, we precisely identify such discontinuities. The arguments stay within the realm of elementary classical analysis and are thus accessible to students encountering a first proof course in the subject.

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International Journal of Mathematical Education in Science and Technology
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