Continuous Function

Date: 05/18/99 at 09:11:55
From: steve
Subject: Abstract analysis

I am interested to know if there is any way of constructing a 
continuous function that is nowhere differentiable. Could you give me 
an example of one such function?

Date: 05/18/99 at 09:38:24
From: Doctor Mitteldorf
Subject: Re: Abstract analysis

Dear Steve,

If you look in an introduction to Wavelet Theory, I think you'll find 
an abundance of such functions and their uses. Here's an example. 
 
On the domain 0 <= x <= 1, start with step 0, y = 0. Next step, create 
a wedge in the middle 1/2, lifting the midpoint by 1 unit

0) y = 0
1) y = 0  for x <= 1/4
   y = 4(x-1/4) for 1/4 <= x <= 1/2
   y = 4(3/4-x) for 1/2 <= x >= 3/4
   y = 0  for x >= 3/4

For the next step, raise the midpoints of each of the four segments by 
1/4 units, again dividing each of the 4 segments into 4.

Continue this process ad infinitum, and you have a continuous function 
of x that is nowhere differentiable.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/