0

(a) Suppose that a function $f:R→R$ is continuous and “vanishes at infinity,” i.e., $lim_{x \rightarrow \infty}f(x) = 0 = lim_{x\rightarrow -\infty} f(x)$. Prove that $f$ is uniformly continuous on $\mathbb{R}$.

(b) Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and there is some $R >0$ such that $f(x) = 0$ whenever $|x|> R$. Prove that $f$ is uniformly continuous on $\mathbb{R}$. (Hint: Even though the domain of $f$ is not a compact set, a theorem from class will be very helpful)

I know that a function $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is uniformly continuous if for any $\epsilon >0$, there exists a $\delta >0$ such that for any $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^n$, \begin{equation*} ||x-y||<\delta \implies ||f(x)-f(y)||<\epsilon \end{equation*}

I am not quite sure how to prove uniform continuity for these 2 parts given this definition. I am thinking the hint in part b refers to the theorem "if f is continuous and $K \subseteq R^n$ is compact, then f is uniformly continuous on the set K.

José Carlos Santos
  • 427,504
Housefire
  • 432
  • You are right about the hint for part b. – Gary Moon Apr 27 '21 at 22:46
  • You are right about the hint for (b); consider the set $[-R,R]$. I would do (b) first and then combine it with the fact that a function that vanishes at infinity must have $|f(x)-f(y)|$ small when both $x$ and $y$ are large enough in absolute value. – Brian M. Scott Apr 27 '21 at 22:47
  • The proof here can be adapted for a) and b) is obvious from a): https://math.stackexchange.com/questions/1145847/f0-infty-rightarrow-mathbbr-is-continuous-prove-uniform-continuity-on?rq=1 – Kavi Rama Murthy Apr 27 '21 at 23:17

1 Answers1

0

For part b, the result result you mentioned is the way to go. It implies that $f$ is uniformly continuous on $\{|x| \leq R\}$. Of course, since $f\equiv 0$ on $\{|x|>R\}$, it is uniformly continuous there as well. Could you figure out how to finish it from there?

For the first part, see this post: A function vanishing at infinity is uniformly continuous. Once you prove part a, part b is just an immediate corollary. That would be an alternative to the above.

Gary Moon
  • 2,495