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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- Nov 25th 2008, 05:45 AMjkruUniform 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. - Nov 25th 2008, 11:56 AMOpalg
Hint: If a function has a bounded derivative then the function is uniformly continuous.

- Nov 25th 2008, 01:15 PMjkru
We're not allowed to use derivatives, unfortunately. Any other ideas on how to attack this problem?

- Nov 26th 2008, 12:19 AMOpalg
In that case, you'll have to use algebra (if that's allowed (Itwasntme) ):

$\displaystyle \left|\frac1{1+x^2} - \frac1{1+y^2}\right| =\frac{|y^2-x^2|}{(1+x^2)(1+y^2)} \leqslant \frac{|x|+|y|}{(1+x^2)(1+y^2)}|y-x|\leqslant |y-x|$, because $\displaystyle \frac{|x|}{1+x^2}\leqslant\tfrac12$. - Nov 26th 2008, 04:51 AMjkru
Haha, yea, I believe it is. Thanks a lot.