Exercise 4.8

The quickest solution uses Exercise 13 below. Note that the closure $Ē$ is also bounded, since if $E\subset [m,M]$ and $x>M$, then $x$ has a neighborhood which doesn’t intersect $E$ so $x\notin Ē$, similarly $y\notin Ē$ if $y<m$, so that $Ē\subset [m,M]$ also. Hence $Ē$ is compact. By Exercise 13, $f$ can be extended to a continuous function $\bar{f}$ on $Ē$ whose range is also compact. Hence $\bar{f}$ is bounded on $Ē$, so $f$ is bounded on $E$.

The identity function $f(x)=x$ on $E=R$ (which is not bounded) is uniformly continuous but not bounded on $E$.