= Solution
Apply <Arzelà-Ascoli theorem> on $[-m,m]$. Pointwise boundedness and equicontinuity give a subsequence converging <uniform convergence>[uniformly] there. Successively extract subsequences for $m=1,2,\ldots$, and take the <diagonal argument>[diagonal subsequence]. It converges uniformly on every closed bounded interval. Its limits on nested intervals agree, so they define $f$ on the <real numbers>; the locally uniform limit of <continuous functions> is continuous.
Back to article page
