From: "G. A. Edgar" <edgar@math.ohio-state.edu.nospam>
Subject: Re: ISO f(x) continuous everywhere, differentiable nowhere
Date: Wed, 17 Mar 1999 09:32:18 -0500
Newsgroups: sci.math
To: Rick Tony <rtony@telerama.com>

[[ This message was both posted and mailed: see
   the "To," "Cc," and "Newsgroups" headers for details. ]]

In article <36EC6BB1.62F0C27A@telerama.com>, Rick Tony <rtony@telerama.com> 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)