On a graph what ways can it not be differentiable? I know at cusps it's not differentiable but are there other instances where it is not differentiable?

Answer 1

in general, it has to be continous at the point, and it has to have the same derivative approaching from the right, and from the left.

not continuous, but has the same derivative:
y= 3 for x<2
y= 5 for x>2
not continous, and does not have the same derivative.
y=-x for x<0
y=x for x>0

and a few folks allow functions that have multi values (shame on them), so that at a particular x, y can have more than one value, take a circle about the origin for example. It is better to not define these as functions, but I know some present these as functions, they are not differentiable unless one specifies exactly which part of it.