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Thread: Derivative Increasing ==> Derivative Continuous

  1. #1
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    Derivative Increasing ==> Derivative Continuous

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

    Ok since $\displaystyle f$ is diff. on $\displaystyle [a,b]$ it's also continuous on $\displaystyle [a,b]$. Since it's increasing we have $\displaystyle f'(x) \ge 0 $ for any $\displaystyle x \in (a,b)$. The first two satisfy the MVT for derivatives so how do I use the MVT and the fact that $\displaystyle f'$ is increasing to show it's derivative is continous? I struggle implementing MVT into my proofs apparently.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ABigSmile View Post
    Prove if $\displaystyle f'$ is increasing on $\displaystyle (a,b)$, . Since it's increasing we have $\displaystyle f'(x) \ge 0 $ for any $\displaystyle x \in (a,b)$.
    Read that again.
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  3. #3
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    Quote Originally Posted by ABigSmile View Post
    $\displaystyle f $ is differentiable on $\displaystyle [a,b]$. Prove if $\displaystyle f'$ is increasing on $\displaystyle (a,b)$, then $\displaystyle f'$ is cont. on $\displaystyle (a,b)$.

    Ok since $\displaystyle f$ is diff. on $\displaystyle [a,b]$ it's also continuous on $\displaystyle [a,b]$. Since it's increasing we have $\displaystyle f'(x) \ge 0 $ for any $\displaystyle x \in (a,b)$. The first two satisfy the MVT for derivatives so how do I use the MVT and the fact that $\displaystyle f'$ 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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