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Math Help - Uniform Continuity

  1. #1
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    Uniform Continuity

    Show that the function f(x) = 1/(1 + x^2) for all x in the Reals is uniformly continuous on the Reals.

    I'm trying to prove this using the definition of uniform continuity, but I'm having some trouble.
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  2. #2
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    Hint: If a function has a bounded derivative then the function is uniformly continuous.
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  3. #3
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    We're not allowed to use derivatives, unfortunately. Any other ideas on how to attack this problem?
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  4. #4
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    In that case, you'll have to use algebra (if that's allowed ):

    \left|\frac1{1+x^2} - \frac1{1+y^2}\right| =\frac{|y^2-x^2|}{(1+x^2)(1+y^2)} \leqslant \frac{|x|+|y|}{(1+x^2)(1+y^2)}|y-x|\leqslant |y-x|, because \frac{|x|}{1+x^2}\leqslant\tfrac12.
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  5. #5
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    Haha, yea, I believe it is. Thanks a lot.
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