From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Differentiable/Continuous problem Date: 21 Jan 2000 02:50:02 -0500 Newsgroups: sci.math Summary: [missing] In article <8686ij$tbm$1@nnrp1.deja.com>, *Paragon* wrote: :Here's a problem given to me, which I thought would be easy at first, :but it turns out to be no so trivial (for me anyway): : :Is there a function f with domain [a,b] so that f is differentiable at :each point in [a,b] but f' is not continuous? This can be interpreted in more than one way: if you want discontinuity of f' at a selected point c, it can be done (and it is a standard example): f(c) = 0, for x other than c, f(x) = (x-c)^2 * sin(1/(x-c)) If you want an unbounded discontinuity of f' at c, replace (x-c)^2 by abs(x-c)^(3/2). If you want f' discontinuous at a finite number of points, just add up examples of the previous type. If you want f' discontinuous everywhere, it cannot be done. The reason is that an everywhere defined derivative is a function of first Baire class (a pointwise limit of a sequence of continuous functions), and such functions are continuous at "most" points of the interval (on a set of second Baire category). If the terms are unfamiliar to you, look them up in a math dictionary. The precise extent to which an everywhere defined derivative can be discontinuous is not known to me. What is known is that the discontinuities cannot be jumps; an everywhere defined derivative has the Intermediate Value Property. Hope it helps, ZVK(Slavek). ============================================================================== From: Simon Subject: Re: Differentiable/Continuous problem Date: Fri, 21 Jan 2000 12:56:27 GMT Newsgroups: sci.math The set of possible discontinuities of an everywhere defined derivative is known, as are the answers to a whole bunch of related problems. See Andrew Bruckner's books for a decent account of this. [quote of previous article deleted --djr] -- Dr Simon Fitzpatrick, Shenton Park, Western Australia Mathematician and International Correspondence Chess Master http://www.q-net.net.au/~dsf/Simon.html