On a discontinuity of a derivative

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 lim_x-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.