polynomials converging point wise to $f$ on $\mathbb{R}$

Question

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous. Would there exist a sequence of polynomials converging point wise to $f$ on $\mathbb{R}$?

I know that it is true on a compact set in $\mathbb{R}$.

Answer 1

By the Weierstrass theorem, for every $k>0$ there is a polynomial approximating $f$ with uniform error bound $1/k$ on $[-k,k].$ Call that polynomial $P_k$. I claim the sequence $P_k$ converges pointwise to $f$. Given $x$, for all $k>|x|$ we have $|f(x)-P_k(x)|\le 1/k$, so for that $x$, $P_k(x)\to f(x)$. But of course this convergence is far from uniform: for any given $k$, the error $P_k(x)-f(x)$ is completely uncontrolled when $|x|>k$.