Can a derivative function have a jump discontinuity at a point and the function still be differentiable at that point? I'm pretty sure it cannot, since it means that the secant slopes do not tend to the same number, but I am not sure how to prove it from the definitions. I have read that the derivative function can have an essential discontinuity at a point and the function still be differentiable at that point. i.e.
$$ f(x) = \left\{ \begin{array}{ll} x^2sin\left(\frac{1}{x}\right) & \quad x \ne 0 \\ 0 & \quad x = 0 \end{array} \right. $$.
Are there any other kinds of discontinuity or are these only the two cases. Many thanks.
It follows from Darboux's theorem, that if $f$ is a differentiable function and if $f'$ is discontinuous at $a\in D_f$, then the only possible cause of that discontinuity is that the limit $\lim_{x\to a}f'(x)$ doesn't exist.