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Math Help - Continuity of a derivative

  1. #1
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    Continuity of a derivative

    \text{Suppose } a, c \in \mathbb{R}, c > 0, f \in C[-1, 1],
    f(x) = \begin{cases} x^a sin(|x|^{-c}) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}

    \text{Show } f \in C^{1}[-1, 1] \iff a > 1 + c.

    What do I do here? I just did a proof that showed that f is bounded iff a => 1 + c (and I had a tough time with that). Can I use that result to show continuity whenever a > 1 + c, or is there more to it? And what to do about the "iff"?

    Please, please, please be clear and explicit in your replies--I am still a rookie at this stuff.
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  2. #2
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    Re: Continuity of a derivative

    Have you at least done the calculations? if x> 0 then f'(x)= ax^{a-1}sin(|x|^{-c})- cx^{a+ c-1} cos(|x|^c). If x< 0 then f'(x)= ax^{a-1}sin(|x|^{-c})+ncx^{a+ c- 1}cos(|x|^{-c}). And f'(0)= 0 as long as a is not 0. Are the limit from above and below equal to 0?
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