Is it correct to say that if $f(x)$ is uniformly continuous on $(-\infty,-1]$ and $[-1,\infty)$, then it is uniformly continuous on $\mathbb{R}$?
I don't think this is true but cannot think of a counterexample. Below there is an example of where I want to use this.
Thanks for any help
Prove that $f(x)=|x|^{\frac{1}{2}}$ is uniformly continuous on $\mathbb{R}$
Proof. As $|x|^{\frac 12}$ is differentiable on $(-\infty,-1]$ and $[1,\infty)$ with the derivatives bounded then it is uniformly continuous on these intervals.
It is also continuous on $[-1,1]$ and so it is uniformly continuous.
It is therefore continuous on $\mathbb{R}$.