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Math Help - continous function

  1. #1
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    continous function

    f(x)=e-1/x2 if x is not =0
    =0 if x=0

    then f is continuous. how it can be proved.
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  2. #2
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    Re: continous function

    Quote Originally Posted by saravananbs View Post
    f(x)=e-1/x2 if x is not =0
    =0 if x=0

    then f is continuous. how it can be proved.
    See if \displaystyle \begin{align*} \lim_{x \to 0^-}e^{-\frac{1}{x^2}} = \lim_{x \to 0^+}e^{-\frac{1}{x^2}} = 0 \end{align*}.
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    Re: continous function

    Quote Originally Posted by saravananbs View Post
    f(x)=e-1/x2 if x is not =0
    =0 if x=0

    then f is continuous. how it can be proved.
    We need to show that \lim_{x\to0}f(x) = f(0).

    \lim_{x\to0}f(x)

    =\lim_{x\to0}e^{-1/x^2}

    Now evaluate the limit and show that it equals f(0).
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  4. #4
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    Re: continous function

    yes at x=0, it is continuous

    but for other points , how its derived.

    is it like limit x-> a+h f(x) = limit x-> a-h f(x) =f(a)?
    Last edited by saravananbs; June 13th 2012 at 08:45 AM.
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  5. #5
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    Re: continous function

    Quote Originally Posted by saravananbs View Post
    yes at x=0, it is continuous

    but for other points , how its derived.
    One way is to say that e^{-1/x^2} is continuous as the composition of continuous functions.
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    Re: continous function

    Quote Originally Posted by emakarov View Post
    One way is to say that e^{-1/x^2} is continuous as the composition of continuous functions.
    i think three function are involved , e^x ,1/x, x^2

    all are continuous. is it right.
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  7. #7
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    Re: continous function

    All are continuous where x =/= 0.
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  8. #8
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    Re: continous function

    Yes, all are continuous for x ≠ 0.
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