1. ## 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.

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

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

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

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