Proving whether $f(x) =|x|^{1/2}$ is uniformly continuous for $f:\mathbb{R}\to\mathbb{R}$ I need to prove following question:
proving whether or not, $f(x) =|x|^{1/2}$ is uniformly continuous for $f:\mathbb{R}\to\mathbb{R}$
My attempt was based on this is not uniformly continuous. following definition and With taking $y=x/2$ at the result I could find some epsilon not providing the definition. I could prove that is continuous but I still suspect about .my attempt and I need hand. 

Given $\epsilon >0$, there is $r>0$ such that $ |f(x)| < \epsilon$ for all $|x|<r$ there.
Outside that neighborhood of origin, say on $|x|>r/2$ the derivative is bounded, thus by mean value theorem uniformly continuous. Patching them together, we get uniform continuity of $f$ on all $\mathbb{R}$.
