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### 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/
```
Associated Topics:
College Analysis

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