Math Help - Strictly Increasing Discontinuous Derivative

1. Strictly Increasing Discontinuous Derivative

Does there exist a function $f:\mathbb{R} \to (a,b)$ whose derivative exists everywhere, is strictly increasing and discontinuous?

Edit:
I'm asking that the derivative be strictly increasing and discontinuous, not $f$

2. Re: Strictly Increasing Discontinuous Derivative

Originally Posted by SlipEternal
Does there exist a function $f:\mathbb{R} \to (a,b)$ whose derivative exists everywhere, is strictly increasing and discontinuous?

Edit:
I'm asking that the derivative be strictly increasing and discontinuous, not $f$
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)