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Thread: disprove uniform continuety

  1. #1
    MHF Contributor
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    disprove uniform continuety

    i need to prove that $\displaystyle \frac{1}{\sqrt{x}}$ is not uniformly continues in (0,1)



    for epsilon=0.5

    $\displaystyle |\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{y}}|=|$$\displaystyle ]\frac{\sqrt{y}-\sqrt{x}}{\sqrt{xy}}\frac{\sqrt{y}+\sqrt{x}}{\sqrt {y}+\sqrt{x}}|$$\displaystyle =|\frac{y-x}{(\sqrt{y}-\sqrt{x})\sqrt{xy}}|$



    i need to prove that the above exprseesion bigger then 0.5



    but i dont know what x and y to choose

    ?
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  2. #2
    Super Member girdav's Avatar
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    Re: disprove uniform continuety

    Consider for each integer $\displaystyle n$: $\displaystyle x_n:=\frac 1n$ and $\displaystyle y_n=\frac 1{2n}$. What about $\displaystyle |x_n-y_n|$ and $\displaystyle \left|\frac 1{\sqrt{x_n}}-\frac 1{\sqrt{y_n}}\right|$?
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