# Strictly Increasing Discontinuous Derivative

• October 11th 2011, 04:40 AM
SlipEternal
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 $f$
• October 11th 2011, 09:05 AM
Drexel28
Re: Strictly Increasing Discontinuous Derivative
Quote:

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)