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Math Help - differentiability proof

  1. #1
    Junior Member
    Joined
    Oct 2009
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    39

    differentiability proof

    a)
    Suppose f(x)=xg(x) for some function g which is continuous at 0. Prove that f is differentiable at 0 and find f'(0) in terms of g.

    b)
    Suppose f is differentiable at 0 and that f(0)=0, prove that f(x)=xg(x) for some function g which is continuous at 0. Hint: what happens if you try to write g(x)=f(x)/x ?
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  2. #2
    Senior Member
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    Feb 2010
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    1) f(h) = h g(h) = h(g(0) + o(1)) = f(0) + hg(0) + o(h) so rearranging gives {f(h) - f(0)\over h} = g(0) + o(1), or sending h\to 0, f'(0) = g(0).

    2) It comes down to showing that the limiting ratio f(x)\over x exists. Use l'Hopitals rule. Note that the result is in agreement with (1), which is always good!
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