TITLE: If $ f \in C_0^\infty$, then is $f$ uniformly continuous?
QUESTION [5 upvotes]: If $ f \in C_0^\infty=\{ g: g\in C^\infty, \lim_{|x|\rightarrow \infty}g(x)=0\}$, then is $f$ uniformly continuous on $\mathbb R$?
($ f : \mathbb R \to \mathbb R $)

REPLY [5 votes]: HINTs

A continuous function on a compact interval is uniformly continuous.
$\lim_{|x| \to \infty} f(x) = 0$ means that $\forall \epsilon...$
Split up the domain to use these two properties.