# Thread: Proving a function is uniformly continuous on R

1. ## Proving a function is uniformly continuous on R

I need to show that the function f(x)=|x|^1/2 is uniformly continuous on R. I know how to start it's just I get confused when it gets to f(x)-f(y). Any help would be appreciated.

2. Originally Posted by lenny
I need to show that the function f(x)=|x|^1/2 is uniformly continuous on R. I know how to start it's just I get confused when it gets to f(x)-f(y). Any help would be appreciated.
What have you tried? What can you use?

If you're allowed to use anything I'd note that on $[-1,1]$ $f$ is continuous and thus by the Heine-Cantor theorem uniformly continuous. Then, for $x\in\mathbb{R}-[-1,1]$ I'd note that $f'(x)$ is bounded and thus $f$ is Lipschitz and so evidently unif. cont.