Uniform Continuity of $\sqrt{x}$

Question

I want to show $\sqrt{x}$ is uniformly continuous on $[0, \infty)$. I know it is uniformly continuous on $[0,1]$ and I can show it's uniformly cts. on $[1, \infty)$, so if I choose $x=1/2$ and $y\in [1,\infty)$, is it uniformly continuous on $[0,\infty)$?

Answer 1

Use the fact that $$|\sqrt{|x|} -\sqrt{|y|} |\leqslant \sqrt{|x-y|}$$ for any $x,y\in\mathbb{R} .$