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Math Help - 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
    f(x)=xcos(1/x), otherwise
    How do i prove that f is continuous at the point a=0?

    I tried to use the \lim_{x\to0}f(x)=f(0) but I can't go any further from \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 " \lim_{x\to 0} cos(1/x)" (which does not exist anyway)? The function you are concerned with is x cos(1/x).

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

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