Is $\sqrt{|x|}$ uniformly continuous from $\mathbb{R}\to\mathbb{R}$?

Question

Is $\sqrt{|x|}$ uniformly continuous from $\mathbb{R}\to\mathbb{R}$?

I can see it is continuous and by the derivative test it is not lipschitz but I can not prove (or disprove) uniformity.

Answer 1

Hint

If $x,y\geq 0$, prove that:

$$|\sqrt{x}-\sqrt{y}|\leq\sqrt{|x-y|}.$$

I let you adapt it when $x,y\in\mathbb R$.