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.
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
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?
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.