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 $)
If $ f \in C_0^\infty$, then is $f$ uniformly continuous?
5
$\begingroup$
real-analysis
-
1I do not think that this is a duplicate of that question. – 2012-06-10
1 Answers
5
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.