From: "G. A. Edgar" Subject: Re: ISO f(x) continuous everywhere, differentiable nowhere Date: Wed, 17 Mar 1999 09:32:18 -0500 Newsgroups: sci.math To: Rick Tony [[ This message was both posted and mailed: see the "To," "Cc," and "Newsgroups" headers for details. ]] In article <36EC6BB1.62F0C27A@telerama.com>, Rick Tony wrote: > happy pi day everyone (3/14). I know there is a function that is > continuous but 'infinitely bumpy' so that its derivative fails at every > point. I know weirstrauss found it a while back, but i can't remember or > find the function itself. As i recall it involves an infinite sum of > cosine functions. Anyone that can help me out with a text reference or > the explicit f(x) would be appreciated. Please email as i don't read the > newsgroup often. > Thanks, Here is something from 1996... It deals with Riemann's example rather than Weierstrass's. > From: wcw@math.psu.edu (William C Waterhouse) > Newsgroups: sci.math > Subject: Re: Continuous and non-differentiable fn' > Date: 16 Aug 1996 21:44:52 GMT > Organization: Dept. of Mathematics, Penn State > Message-ID: <4v2q4k$49s@dodgson.math.psu.edu> > > A week or so ago, I posted the following information about the > function > > f(x) = Sum[n=1 to inf] sin(n^2 * pi * x) /n^2: > > "This question was settled by Joseph Gerver in two papers in the > American Journal of Mathematics, 92 (1970) 33-55 and > 93 (1971), 33-41. The function has derivative -1/2 at all > points x = (2m+1)/(2n+1) and is nondifferentiable elsewhere." > > Several people, e.g. Don Redmond, have now mentioned work of A. Smith > on the problem, and I have tracked that down. > > 1) The paper in question was in Proc. Amer. Math. Soc. 34 (1972), > 463-468. > > 2) It was written when he knew Gerver's first paper (the > differentiability > at certain rationals) but before he saw Gerver's second paper. > > 3) It has a simpler method. > > 4) It also shows that there are infinite derivatives at all rationals > where there is no finite derivative (and not even an infinite > derivative for irrational x). > > In summary, the priority belongs to Gerver, but Smith's paper is > the one to read now. > > I should add that I also located a discussion of this topic > by S. L. Segal, Math. Intelligencer 1 (1977), no. 2, 81-82. > > William C. Waterhouse > Penn State -- Gerald A. Edgar edgar@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax)