Is it possible to uniformly approximate every continuous function $f: \mathbf{R} \rightarrow \mathbf{R}$ by smooth functions?
In other words, is it true that for each continuous function $f: \mathbf{R} \rightarrow \mathbf{R}$ there exists a sequence $(f_n)$ of smooth functions $f_n: \mathbf{R} \rightarrow \mathbf{R}$ such that $f_n \rightarrow f$, as $n\rightarrow \infty$, uniformly on $\mathbf{R}$?
Thanks.