Real analysis. Uniformly continuous

Question

Suppose $$f:\mathbb R\to\mathbb C$$ is continuous and $f(x)=0$ for all $|x|>1$.

Show $f$ is uniformly continuous on $\mathbb R$.

This is not homework. I'm trying to study for a test. I appreciate the help.

Answer 1

Some hints:

1. Show that $f$ is uniformly continuous on $[-2,2]$ by the theorem of your choice.
2. For a given $\varepsilon$, the $\delta<1$ obtained in 1. such that $|x-y|\lt \delta, x,y\in [-2,2]$ implies $|f(x)-f(y)|\lt\varepsilon$ actually works for $x,y$ in the whole real line.