Does the Stone-Weierstrass theorem (Weierstrass approximation theorem) apply in $\mathbb{R}$.
The Stone-Weierstrass theorem in $\mathbb{R}$ would be:
Given continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ and $\epsilon>0$, there exists a polynomial $p(x):\mathbb{R}\rightarrow\mathbb{R}$ such that $|f(x)-p(x)|<\epsilon$ for all $x\in\mathbb{R}$.