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Math Help - Derivative Increasing ==> Derivative Continuous

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    Derivative Increasing ==> Derivative Continuous

    f is differentiable on [a,b]. Prove if f' is increasing on (a,b), then f' is cont. on (a,b).

    Ok since f is diff. on [a,b] it's also continuous on [a,b]. Since it's increasing we have f'(x) \ge 0 for any x \in (a,b). The first two satisfy the MVT for derivatives so how do I use the MVT and the fact that f' is increasing to show it's derivative is continous? I struggle implementing MVT into my proofs apparently.
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    Quote Originally Posted by ABigSmile View Post
    Prove if f' is increasing on (a,b), . Since it's increasing we have f'(x) \ge 0 for any x \in (a,b).
    Read that again.
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    Quote Originally Posted by ABigSmile View Post
    f is differentiable on [a,b]. Prove if f' is increasing on (a,b), then f' is cont. on (a,b).

    Ok since f is diff. on [a,b] it's also continuous on [a,b]. Since it's increasing we have f'(x) \ge 0 for any x \in (a,b). The first two satisfy the MVT for derivatives so how do I use the MVT and the fact that f' is increasing to show it's derivative is continous? I struggle implementing MVT into my proofs apparently.
    If you know the IVT for derivatives then just answer this: What kind of discontinuities can a monotone function have?
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