Does there exist a function $\displaystyle f:\mathbb{R} \to (a,b)$ whose derivative exists everywhere, is strictly increasing and discontinuous?
Edit:
I'm asking that the derivative be strictly increasing and discontinuous, not $\displaystyle f$
Does there exist a function $\displaystyle f:\mathbb{R} \to (a,b)$ whose derivative exists everywhere, is strictly increasing and discontinuous?
Edit:
I'm asking that the derivative be strictly increasing and discontinuous, not $\displaystyle f$
No, there does not. By Darboux's theorem derivatives have the intermediate value property, and it's a common fact that an injective function on an interval that has the intermediate value property is continuous (cf. here for example)