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Math Help - Uniform Continuity

  1. #1
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    Uniform Continuity

    Qn: Show that the function f(x) = 1/(1+x^2) is uniformly continuous on R.

    I tried to show that the function satifies the Lipschitz condition

    |1/(1+x^2) - 1/(1+y^2)| = |(x^2 - y^2)/(1+x^2)(1+y^2)| = |(x+y)/(1+x^2)(1+y^2)||x-y| --- (*)

    Then I observe that since (x-1)^2 >= 0. Hence 1+x^2 >= 2x and 1+y^2 >= 2y.

    Thus (*) <= |(x+y)/4xy||x-y| = 1/4 |1/x + 1/y||x-y|

    However, I am unable to proceed after that. How should i do it? I have also tried to prove it by using the epsilon-delta definition of uniform continuity but the argument is very similar to the one above.

    Thanks.
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  2. #2
    Super Member girdav's Avatar
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    Re: Uniform Continuity

    You only have to show that g(x,y):=\frac{x+y}{(1+x^2)(1+y^2)} is bounded independently of x and y. Write \left|g(x,y)\right| =\left|\frac{x}{(1+x^2)(1+y^2)}\right|+\left|\frac  {y}{(1+x^2)(1+y^2)}\right|\leq \frac{|x|}{1+x^2}+\frac{|y|}{1+y^2}, and x\mapsto \frac{|x|}{1+x^2} is bounded by \frac 12.
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