I was reading a book containing a result which I would summarize (perhaps incorrectly) as stating that differentiable functions cannot have "jump" discontinuities -- where the limit "from the left" is different from the limit from the right.
The actual theorem is:
If f is differentiable with a finite derivative $f'(t)$ in an interval, then at all points $f'(t)$ is either continuous or has a discontinuity of the second kind.
Discontinuity of the second kind is earlier defined to be any discontinuity that is not a jump discontinuity.
The same result (I think) has been proved/discussed here previously. The actual proof, both in the link and in the book, involves using the mean value theorem and the definition of the derivative to show that at all points in the interval the derivatives' limit-from-left is equal to its limit-from-right.
But ... this seems more sophisticated than necessary to me. If f is differentiable in an interval, then is it not true by definition that the derivative cannot have jump discontinuities? For if the derivative had a jump, the function would not actually be differentiable at this point, for example the absolute value function at zero.
I'm wrong here somewhere but can't tell where. Help appreciated.
Edit: changed phrasing of my objection immediately above to clarify what I mean.
