Expert: Paul Klarreich - 12/9/2006

Question
1. is a continous func always monotonic in some sub domain?
2.  is there a function which is continous everywhere but      differentiable nowhere?

Answer
Questioner:  suraj
 
Subject:  functions
Question:  1. is a continous func always monotonic in some sub domain?
2.  is there a function which is continous everywhere but differentiable nowhere?

Hi, Suraj,

I believe the answers to these questions are the same function, which I will try to construct:

The function will be the limit of the following iterative process:

1. Start with a 'sawtooth function'.  Sort of what you would get if you tried to make a sine curve but ran out of curved pieces and had to do it with straight pieces:

/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\

2. Now look at one of the segments, which is a line segment of the form:

y = mx + n,  between  xa and xb.

So it is a line segment between (xa,ya) and (xb,yb).  Let's assume it's one of the pieces that go 'up', that is, yb > ya.

WARNING: THE FOLLOWING DISCUSSION MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  BE SURE TO VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.

So it looks like:

       /
      /
     /
    /
   /
  /
 /
/
/
xa      xb

Divide the interval between xa and xb into five equal pieces and let  xa1, xa2, xa3, xa4 be the points of division.  Erase the middle three pieces and replace them with the pieces that join new values of  ya2 and  ya3 so that:

A.  ya2 < ya1, so the piece from xa1 to xa2 goes down instead of up.
B.  ya3 > ya4, so the piece form xa3 to xa4 goes down instead of up, AND the piece from xa2 to xa3 goes up.


         /
      /\/
     /
    /
   /
/\/
/

So we have these subintervals:
  Up      Down    Up     Down     Up     Down, because:
xa --- xa1 --- xa2 --- xa3 --- xa4 --- xb --->

because, xa --> xb  was originally an 'up'.

Of course, you do this process for each segments.

2. Now iterate this process for each of the new, smaller segments.  Take the limit of this function as the number of iterations approaches infinity.

.................................................
The resulting function (I believe some call this a 'fractal') has two interesting (some might call them disgusting) properties:

A. There is no subinterval over which it is monotonic.

B. Since at every value of x, you have a 'sharp corner' it is not differentiable for any value of x.  Of course, we never 'broke' the function, so it is continuous everywhere.