If derivative of $f$ is continuous, then $f$ is continuous. I have searched a lot, but i haven't found any proof about that statement. I have checked the proof of

If $f$ is differentiable, then $f$ is continuous

but it's not the same argument I think. Also, I want to know what's your opinion about the statement

If derivative of $f$ is not continuous, then $f$ is not continuous


If $f$ is differentiable, then $f$ is continuous. The continuity of $f'$ is irrelevant here.
In particular, even if $f'$ is discontinuous, $f$ is continuous.
