# When is a function not differentiable?

I'm curious about the conditions for when a function f(x) is not differentiable

At a point, or cusp.
At a non-removable discontinuity (such as a jump discontinuity).
On an asymptote.
Probably some other places too.

Also at a place where the slope is vertical.

How about any time it fails the definition of the derivative (i.e., the limit in the definition doesn't exist).