I already know that if $E\subset\mathbb{R}$, $E_{1}$ is a dense subset of $E$, and there is a uniformly continuous function $f_{1}(x)$ on $E_{1}$, then there is a unique function $f(x)$ such that $f(x)=f_{1}(x)$ when $x\in E$ and $f(x)$ is continuous on $E$. In particular, when $E$ is bounded, $f(x)$ is uniformly continuous.

Now I wonder if $E$ is unbounded then the uniform continuity of $f(x)$ still holds. For instance, let $E_{1}=\mathbb{R}\backslash\mathbb{Q}$ and $E=\mathbb{R}$, then can we have $f(x)$ is uniformly continuous on $\mathbb{R}$?

Any help would be appreciated!

  • $\begingroup$ @Klaus Thanks very much! $\endgroup$
    – mio
    Oct 12, 2022 at 14:23