# Can $f'(c)$ exist if $f'(x)$ is not continuous at c

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.

• The derivative need not be continuous, but it must satisfy the IVP. – Randall Jan 11 at 15:34

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.
• but can f'(a) still exist then? I know intuitively it can't but I am not too sure how to show it. Can the derivative function look like $$f'(x) = \left\{ \begin{array}{ll} -1 & \quad x < 0 \\ 0 & \quad x=0\\ 1 & \quad x > 0 \end{array} \right.$$ – matt Jan 11 at 16:04