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Math Help - Second differentiability

  1. #1
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    Second differentiability

    I'm kinda stuck on this problem. Would someone give me a hand?
    Suppose f:[a,b] \rightarrow R is differentiable and f' is differentiable at t \in (a,b). Show that
    f''(t)=\lim_{h\to 0} \frac{f(t+h)-2f(t)+f(t-h)}{h^2}
    Is there a function that satisfies this limit but f' is not differentiable at t?
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  2. #2
    MHF Contributor chisigma's Avatar
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    For the function...

    f(t) = t^{2}\cdot sgn(t) (1)

    ... in t=0 the limit exists and it is equal to 0 but its derivative is not differenziable in that point...

    Kind regards

    \chi \sigma
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jackie View Post
    I'm kinda stuck on this problem. Would someone give me a hand?
    Suppose f:[a,b] \rightarrow R is differentiable and f' is differentiable at t \in (a,b). Show that
    f''(t)=\lim_{h\to 0} \frac{f(t+h)-2f(t)+f(t-h)}{h^2}
    Is there a function that satisfies this limit but f' is not differentiable at t?
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