I understand at cusps, corners, etc, because the negative and positive directions do not agree with eachother. But what about at jump discontinuity on a graph? Why wouldn't a function be differentiable there? I understand that from the definition of differentiable that it just isn't, but I don't get WHY. Using the epsilon-delta definition, you still can get within delta for any given delta, it's just that it doesn't happen to work at that exact point....but why does it have to? Derivatives are as delta x approaches zero...as long as it is approaching zero, and not actually zero, the derivative still holds up...so f(x)+dx does not equal f(x) even if it is very, very close. Not sure if I worded this properly but maybe somebody will get what I mean?