real analysis - Is this a necessary and sufficient condition for the derivative to exist at $C$?

Suppose we want to prove that the derivative of a function across an interval exists at $C$, but the derivative at $C$ cannot be found. We know the function must be continuous. Can we take the limit of derivative from the negative and positive direction of $C$ and show that if they are equal, the derivative at $C$ exists and is equal to the limit obtained? Is this a necessary and sufficient condition?

EDIT:

Sufficiency - If a function is a derivative along some interval, it does not have a removable singularity at $C$.

Necessity - There is no interval of a derivative of some function in which a jump or essential discontinuity occurs.

There are two cases in which the condition is met if this is a necessary and sufficient condition. One is where the derivative is continuous, the other is where there is a removable discontinuity in the derivative. Is the latter possible?