How do I show that $f=x/(1+x^2)$ is uniformly continuous on $\Bbb{R}$
Here is what I did: I took the derivative of $f$ and the $\lim_{x\to\infty}f'(x)$ and found that it goes to $0$. So the derivative of $f$ is bounded.
So since the derivative of $f$ is bounded, $f$ is considered to be Lipschitz. SO ${|f(x)-f(y)|\over|x-y|}< M\text{ for }M>0$
or $|f(x)-f(y)
Please give me some feedback on this?