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Thread: Prove f continuous at a point

  1. #1
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    Prove f continuous at a point

    f:[0,1]--->R given by
    f(x)=0 if x=0
    $\displaystyle f(x)=xcos(1/x)$, otherwise
    How do i prove that f is continuous at the point a=0?

    I tried to use the $\displaystyle \lim_{x\to0}f(x)=f(0)$ but I can't go any further from $\displaystyle \lim_{x\to0}cos(1/x)$. maybe I have to use epsilon/delta definition to solve this?

    thanks
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  2. #2
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    Why "$\displaystyle \lim_{x\to 0} cos(1/x)$" (which does not exist anyway)? The function you are concerned with is $\displaystyle x cos(1/x)$.

    Since $\displaystyle -1\le cos(1/x)\le 1$ no matter what x is, $\displaystyle -x\le x cos(1/x)\le x$ for all x.
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  3. #3
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    Is this correct?

    $\displaystyle \lim_{x\to0}|x{cos{(1/x)}}|\le\lim_{x\to0}|x|=f(0)=0$
    Last edited by charikaar; Apr 11th 2010 at 08:40 AM.
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