Associated Topics || Dr. Math Home || Search Dr. Math

### Continuous but Not Differentiable?

```Date: 12/27/2003 at 02:47:40
From: Amir
Subject: Weierstrauss

I have heard that Weierstrauss found a function,

f(x) = sigma(b^n*cos(a^n*pi*x))

which is continuous everywhere, but differentiable nowhere.  How is
that possible?

```

```
Date: 12/29/2003 at 14:27:07
From: Doctor Fenton
Subject: Re: Weierstrauss

Hi Amir,

Thanks for writing to Dr. Math.  Weierstrass's example is quite
complicated, and although searching the web with the phrase "nowhere
differentiable" found a number of sites which state examples or give
graphs or animations, I found no site which actually proves the
statement.

T. W. Koerner, in his book _Fourier Analysis_, gives a detailed proof
of the non-differentiability of the function

oo
---
\    sin((k!)^2 t)
f(t) = /    -------------
---        k!              .
k=0

Analyzing the difference quotient

oo
---
f(t+h) - f(t)     \    1 [ sin((k!)^2(t+h)) - sin((k!)^2 t) ]
-------------  =  /    - [ -------------------------------- ]
h           ---  h [               k!                 ] ,
k=0

there is a single term in the series whose value dominates the other
terms in the series.  However, this is a fairly complicated argument,
and I would refer you to Koerner's book.  He says that using the same
technique, one can prove the non-differentiability of the Weierstrass
function.

If you just want an example of a non-differentiable function, van der
Waerden's example is much simpler to analyze.

Let {x} denote the "distance to the nearest integer function":
on [0,1),

{   x    0 <= x <= .5
{x} = {
{ 1 - x  .5 < x <  1     ,

and {x} is periodic with period 1 for other values of x.

Then define

oo
---
\    {10^m x}
g(x) = /    --------
---    10^m       .
m=0

Because of the periodicity of {x}, we need consider only points in
[0,1).  Let a be a point in [0,1), with decimal expansion

a = 0.a(1)a(2)a(3)...  .

Define a sequence h(n) by

{  10^(-n)      if a(n) is not 4 or 9;
h(n) = {
{ -10^(-n)      if a(n) is 4 or 9      .

[For example, if a = 0.1497... , then

a(1) = 1, so h(1) = .1   ; a(2) = 4, so h(2) = -.01 ;
a(3) = 9, so h(3) = -.001; a(4) = 7, so h(4) = .0001; etc. ]

In the difference quotient

oo
---
g(a+h(n)) - g(a)     1  \   {10^m (a+h(n))} - {10^m a}
---------------- = ---- /   --------------------------
h(n)         h(n) ---         10^m
m=0

the right side reduces to the sum of n terms, each of which is +1 or
-1, so it is an integer of the same parity as n.  As n->oo, the
difference quotients are integers which change between even and odd
with each increase of 1 in n, so they cannot possibly converge, and g
is not differentiable at a.

If you have any questions, please write back and I will try to explain
further.

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

```
Date: 12/29/2003 at 23:42:18
From: Amir
Subject: Thank you (Weierstrauss)

Thank you for your help!  I did not think that you would answer my
question so immediately.  Your site will be very good for me and many
others.

Once again, thanks for your help.
```
Associated Topics:
College Analysis
College Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/