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Thread: Absolute continuity

  1. #1
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    Absolute continuity

    How to prove that x^1/3 is absolutely continuous without using
    the fact that it is the integral of a derivative.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Absolute continuity

    Quote Originally Posted by hgd7833 View Post
    How to prove that x^1/3 is absolutely continuous without using
    the fact that it is the integral of a derivative.
    I remember this being annoying to do. You know everything's fine on any union of intervals bounded away from zero, since $\displaystyle x^{\frac{1}{3}}$ has bounded derivative, and thus is Lipschitz. Try then to take care of the case of finite intervals containing $\displaystyle 0$. Is this where you're stuck?
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  3. #3
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    Re: Absolute continuity

    I would like to prove that f is absolute continuous over the closed interval [-1,1] .
    Actually the derivative is unbounded there thus not Lipschitz.
    I am trying to use the definition.
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