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Thread: Uniform Continuity for 1/x

  1. #1
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    Uniform Continuity for 1/x

    I'm trying to prove the following statement:

    Prove that the function f(x)=1/x is continuous on (0,1] but it is not uniformly continuous on this interval.

    Thanks a lot.
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  2. #2
    Senior Member Tinyboss's Avatar
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    For the "not uniformly continuous" part: do you know how to formally negate a logical proposition, i.e. exchange "for-all's" and "there-exists's" and reverse (in)equalities? Try that with the definition of uniform continuity.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by AKTilted View Post
    I'm trying to prove the following statement:

    Prove that the function f(x)=1/x is continuous on (0,1] but it is not uniformly continuous on this interval.

    Thanks a lot.
    I actually discuss this in my post here. In essence, (we'll speak less generally now) if $\displaystyle f(0,1]\to\mathbb{R}$ were uniformly continuous, then we could extend it to some uniformly continuous map $\displaystyle \tilde{f}:[0,1]\to\mathbb{R}$. But, this is just nonsense since we'd have to have that $\displaystyle f(0)=\lim f\left(\frac{1}{n}\right)=\lim n$
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by AKTilted View Post
    I'm trying to prove the following statement:

    Prove that the function f(x)=1/x is continuous on (0,1] but it is not uniformly continuous on this interval.

    Thanks a lot.

    The negation of uniform continuity:

    For some $\displaystyle \epsilon > 0 $ and for each $\displaystyle \delta > 0$
    there exist $\displaystyle x_0, x_1$ such that $\displaystyle |x_0 - x_1| < \delta $ but $\displaystyle |f(x_0) - f(x_1)| \geq \epsilon$.

    Assume that there is exist$\displaystyle \delta <\frac{1}{n}$ $\displaystyle n\in \left \{\mathbb{N} / 0}{ \right \}$.

    Now let $\displaystyle x_0,x_1 \in\mathbb{R}^+ x_0=x_1+\frac{\delta }{2} and x_1=\frac{\delta }{2}$


    $\displaystyle \left | f(x_0)-f(x_1) \right |=\left | \frac{1}{x_0}-\frac{1}{x_1} \right |=\left |\frac{ x_0-x_1}{x_0x_1} \right |
    $

    $\displaystyle =\frac{\frac{2}{n}}{x_0(x_0+\frac{\delta }{2})}\frac{\delta }{2}=\frac{\frac{2}{n}}{\delta^2}\frac{\delta }{2}=\frac{\frac{1}{n}}{\delta }>1$
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    Another much better solution:


    Choose $\displaystyle x_n=\frac{1}{n} $and $\displaystyle y_n=\frac{1}{n+1} $for any $\displaystyle n\in\mathbb{N}$.


    $\displaystyle \left | x_n-y_n \right |<\frac{1}{n}$, but $\displaystyle \left | f(x_n)-f(y_n) \right |=1
    $.
    Last edited by Also sprach Zarathustra; Nov 4th 2010 at 06:29 AM.
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