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 $\displaystyle f$

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- Oct 11th 2011, 04:40 AMSlipEternalStrictly Increasing Discontinuous Derivative
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 $\displaystyle f$ - Oct 11th 2011, 09:05 AMDrexel28Re: Strictly Increasing Discontinuous Derivative
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)