# Uniform Continuity

• Nov 25th 2008, 06:45 AM
jkru
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.
• Nov 25th 2008, 12:56 PM
Opalg
Hint: If a function has a bounded derivative then the function is uniformly continuous.
• Nov 25th 2008, 02:15 PM
jkru
We're not allowed to use derivatives, unfortunately. Any other ideas on how to attack this problem?
• Nov 26th 2008, 01:19 AM
Opalg
In that case, you'll have to use algebra (if that's allowed (Itwasntme) ):

$\left|\frac1{1+x^2} - \frac1{1+y^2}\right| =\frac{|y^2-x^2|}{(1+x^2)(1+y^2)} \leqslant \frac{|x|+|y|}{(1+x^2)(1+y^2)}|y-x|\leqslant |y-x|$, because $\frac{|x|}{1+x^2}\leqslant\tfrac12$.
• Nov 26th 2008, 05:51 AM
jkru
Haha, yea, I believe it is. Thanks a lot.