Stone Weierstrass on noncompact subset I would like to ask whether we can create a function $f:\mathbb{R}\rightarrow\mathbb{R}$ which is continuous on $\mathbb{R}$ but $f$ is not the pointwise limit of any sequence of polynomial function $\{p_n(x)\}_{n\in\mathbb{N}}$.
Thank you for all helping.

There is a nice strengthening of Weierstrass' approximation theorem, due to Torsten Carleman (1927):
If $\epsilon:\Bbb R\to (0,\infty)$ is an arbitrary continuous and positive function, then for each continuous $f:\Bbb R\to \Bbb C$ there is an entire function $h:\Bbb C\to \Bbb C$ such that$$|h(x)-f(x)|<\epsilon(x)$$ for all $x\in \Bbb R$.
That $h$ is entire means that it can be written as a power series $h(z)=\sum_{n=0}^\infty a_n z^n$ which converges uniformly on compact subsets of $\Bbb C$. The function $\epsilon(x)$ may of course be constant, but it can also $\to 0$ as fast as desired as $|x|\to\infty$. So this result is stronger than uniform approximation on a noncompact set!
