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Thread: Finding a function that works

  1. #1
    Newbie
    Joined
    May 2008
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    16

    Finding a function that works

    Hey guys, so i have the following:

    $\displaystyle f''(a) = \lim_{h \to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}$

    and I'm trying to figure out a function that works for that limit, but doesn't have a second derivative at $\displaystyle a$... been scratching my head over this one for a while lol thanks!
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  2. #2
    Senior Member
    Joined
    Feb 2010
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    422
    edit2: wait, no, I was right lol. Let a=0 and try $\displaystyle x^2\sin (\frac1x)$.
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  3. #3
    Junior Member
    Joined
    Apr 2010
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    33
    $\displaystyle f(a)=0$

    $\displaystyle f(x)=1$ if $\displaystyle x<a$

    $\displaystyle f(x)=-1$ if $\displaystyle x>a$

    The limit clearly exists and is zero, for the numerator is identically zero. But $\displaystyle f(x)$ is discontinuous at $\displaystyle a$ and hence has no derivatives there.
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