Uniform Continuity

**jkru** · November 25th 2008, 05:45 AM

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.

**Opalg** · November 25th 2008, 11:56 AM

Hint: If a function has a bounded derivative then the function is uniformly continuous.

**jkru** · November 25th 2008, 01:15 PM

We're not allowed to use derivatives, unfortunately. Any other ideas on how to attack this problem?

**Opalg** · November 26th 2008, 12:19 AM

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 $\frac{|x|}{1+x^2}\leqslant\tfrac12$ .

**jkru** · November 26th 2008, 04:51 AM

Haha, yea, I believe it is. Thanks a lot.

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