This is correct. You cannot have a discontinuity with any accepted definition of a derivative (and your definition is explicit about this: the value f(c) must exist). Just allowing the limits on both sides to be equal already has a mathematical definition which is that of a functional limit, the function in this case being (f(x) - flim(c))/ (x-c) where flim(c) is the value of a (different) functional limit of f(x): x->c (as f(c) doesn't exist).

and yes, by defining a new function with that hole explicitly filled in with a defined value to make it continuous is the typical prescription. It does not imply the derivative exists for the other function as the other post posits.