= Pointwise approximation by uniformly continuous functions
Every continuous $f:\mathbb R\to\mathbb R$ can be approximated with <pointwise convergence> by <uniformly continuous> functions. Set $c_n(x)=\max(-n,\min(x,n))$ and $f_n=f\circ c_n$. The map $c_n$ is <Lipschitz continuous> with constant one and $f|_{[-n,n]}$ is <uniformly continuous> by the <Heine-Cantor theorem>, so $f_n$ is <uniformly continuous> on the whole line. On every fixed bounded interval, $f_n=f$ eventually.
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