I haven't seen this question posed by a book. A proof or counter-example would be great for this statement:

Let f be a function whose derivative f ' exists everywhere on a closed interval [a,b]. Then (so I claim) f ' cannot have a JUMP discontinuity.

I believe this is true, and I included the restriction on the type of discontinuity because there would be a counterexample to the claim that " f ' must be continuous ": consider

f(x) = [ x^2 sin(1/x) ] + x

So my question is simply, if the derivative exists everywhere, can it have infinite or finite jump discontinuities? To define that rigorously, I am asking: is it possible that there can be a point c such that the right-handed and left-handed limits of the derivative at c exist but are not equal?

You may note that the absolute value function fails to be a counterexample because the derivative doesn't exist at 0.