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Math Help - Is this function differentiable?

  1. #1
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    Is this function differentiable?

    Let (piece wise function) f(x)= x^(1/3), for x=/= 0; 0 for x=0.

    So is this function differentiable at x=0? Could you explain why or why not? Thanks!
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  2. #2
    Member Abu-Khalil's Avatar
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    The function differentiable at x=0 iff the limit \lim_{h\to 0}\frac{f(h)-f(0)}{h} exists. But \lim_{h\to 0}\frac{f(h)-f(0)}{h}=\lim_{h\to 0}\frac{h^\frac{1}{3}-0}{h}=\lim_{h\to 0}h^{-\frac{2}{3}}, so the limit doesn't exist.
    Last edited by Abu-Khalil; February 6th 2009 at 08:51 PM.
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