# A Function with no Derivative?

1. Dec 1, 2011

### FeDeX_LaTeX

Hi,

Is there any function defined such that it is non-differentiable at every point?

Of course, cusps and asymptotic graphs aren't differentiable at those points, but what about one that can't be differentiated anywhere? I know there are crazy functions like defining some function to be 0 if x is rational and 1 if x is irrational.

Thanks.

2. Dec 1, 2011

### Office_Shredder

Staff Emeritus
There are in fact functions which are continuous at every point and differentiable nowhere. They're pretty complicated. One example:
http://en.wikipedia.org/wiki/Weierstrass_function

One example that's maybe easier to explain intuitively is Brownian motion. The basic idea is that you have an object that's moving up or down at random. Then f(t) is the height of the object at time t. Then the height of the object is continuous, but over arbitrarily small time intervals it changes whether it's moving up or down, so the derivative is not defined

3. Dec 1, 2011

### FeDeX_LaTeX

Thanks for the reply. I have heard of the Weierstrass function but did not know it was differentiable nowhere. Thanks for this.

I sort of get what you mean by the Brownian motion example.

How would one plot a graph of the function I alluded to, where f(x) = 0 if x is rational and f(x) = 1 if x is irrational?

Last edited: Dec 1, 2011
4. Dec 1, 2011

### Number Nine

You wouldn't. The best you could do is plot a grossly simplified kind-of sort-of approximation with a discrete number of values on a small interval.