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Math Help - Uniform Cont.

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    Uniform Cont.

    Prove that f(x)=\frac{1}{x^2+1} is unif. cont. on \mathbb{R}. Any help is appreciated. The first thing I don't understand is how should I start? Given epsilon > 0. Then I gotta choose the right delta I assume. What confuses me on most of these problems it gives an interval but not on this one.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Math2010 View Post
    Prove that f(x)=\frac{1}{x^2+1} is unif. cont. on \mathbb{R}. Any help is appreciated. The first thing I don't understand is how should I start? Given epsilon > 0. Then I gotta choose the right delta I assume. What confuses me on most of these problems it gives an interval but not on this one.
    \left|\frac{1}{x^2+1}-\frac{1}{y^2+1}\right|=\left|\frac{y^2+1-x^2+1}{\left(x^2+1\right)\left(y^2+1\right)}\right  |=|x-y|\frac{|x+y|}{\left(x^2+1\right)\left(y^2+1\right  )}\leqslant |x-y|. Thus, the function is Lipschitz, so automatically uniformly continuous.
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