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.