real analysis - derivative of a function has only 2nd kind discontinuities

How would I be able to show the following claim

If $f$ is differentiable with a finite derivative in an interval, then at all points $f'(t)$ is either continuous or has a discontinuity of the second kind. By just chasing definitions, I can boil the problem down to whether or not one is able to switch the limits in the following $lim_{s\downarrow t}\lim_{c\rightarrow 0} \frac{f(s+c)-f(s)}{c}$. Any help would be highly appreciated.

Answer

Derivative of a function satisfies Intermediate Value Property so the only discontinuity a derivative can have is of the second kind.