How do I show the uniform continuity of $\tan^{-1}$ over $\mathbb{R}$ How do I show the uniform continuity of $\tan^{-1}$ over $\mathbb{R}$ ? I am trying to use the $\epsilon - \delta$ definition. I have just started learning this topic.

Yet another idea. Use
$$\arctan(x) - \arctan(y) = \arctan\left(\frac{x - y}{1 + xy} \right)$$
and the uniform continuity of $\arctan$ in the compact interval $[-h,h]$, $h>0$.
