When you say a function is not differentiable?

Well, firstly, the derivative of a function simply refers to slope. Usually we say that the function is not differentiable at a point.

Say you have a function such as this:

f(x)=|x|

Another way to represent that would be as a piece-wise function:

g(x) = { -x for x<0

{ x for x>= 0

The problem arises at the specific point x=0. If you look at the slope--the change in the function--from the left and right of x, you notice that it is different, negative 1 and positive 1. So, we can say that the function is not differentiable at x=0 because of that sudden change.

There are however, a few functions that are nowhere differentiable. One example is the Weirstrass function. The even more ironic thing about this function is that it is continuous everywhere! Since this function is not differentiable anywhere, many might call it a non-differentiable function.

There are absolutely other examples.