From: "G. A. Edgar" <edgar@math.ohio-state.edu.nospam>
Subject: Re: How to sum e(i n^2 x)/n^2 ?
Date: Thu, 11 May 2000 08:51:47 -0400
Newsgroups: sci.math
Summary: [missing]

> >
> >    oo
> >   ----     i n^2 x
> >   \       e
> >    |     ----------
> >   /         n^2
> >   ----
> >   n=1
> >
> [...]
> 
>         Don't quote me on this part:

OK, I removed your name... :-)

> I may be remembering
> something wrong, but I tend to suspect that this function
> is simply _not_ differentiable; in fact I think that the
> real part is more or less an incorrect example Riemann gave 
> of a nowhere-differentiable function.

I think Riemann only cloimed that it (actually, its imaginary part)
was non-differentiable at an everywhere-dense set of points.
So I would not call it "incorrect" exactly.

> Riemann's example 
> turned out to be not quite nowhere-differentiable, but it's 
> non-differentiable at most points.

Hardy showed that it is non-differentiable everywhere except at
certain rational values of x.
Trans. Amer. Math. Soc. 17 (1916)

-- 
Gerald A. Edgar              edgar@math.ohio-state.edu