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Math Help - Differentiability Proof of two functions

  1. #1
    Member thaopanda's Avatar
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    Differentiability Proof of two functions

    Let f,g :=  R \rightarrow R be given by

    f(x) :=
    x sin(\frac{1}{x}), x \neq 0
    0, x = 0

    and

    g(x) :=
    x^2 sin(\frac{1}{x}), x \neq 0
    0, x = 0

    Prove that f is not differentiable at x = 0 and that g is differentiable at x = 0.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by thaopanda View Post
    Let f,g :=  R \rightarrow R be given by

    f(x) :=
    x sin(\frac{1}{x}), x \neq 0
    0, x = 0

    and

    g(x) :=
    x^2 sin(\frac{1}{x}), x \neq 0
    0, x = 0

    Prove that f is not differentiable at x = 0 and that g is differentiable at x = 0.
    A function F(x), is differentiable at x= 0 if and only if the limit \lim_{h\to 0}frac{F(h)- F(0)}{h} exists. Do that for both f(x) and g(x) and see what you get!
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