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.
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