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Math Help - Strictly Increasing Discontinuous Derivative

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    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
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Strictly Increasing Discontinuous Derivative

    Quote Originally Posted by SlipEternal View Post
    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)
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