The set of pointwise limits of continuous functions from from $\mathbb{R}$ to $\mathbb{R}$ is the set of Baire class 1 functions. My question is, my question is, what is the set of pointwise limits of polynomials from $\mathbb{R}$ to $\mathbb{R}$?
The Weierstrauss approximation theorem implies that every continuous function is a pointwise limit of polynomials. But are there also discontinuous functions which are pointwise limits of polynomials?
Best Answer
Any Baire class 1 function is a pointwise limit of a sequence of polynomials. Indeed, let $f:\mathbb{R}\to\mathbb{R}$ be of Baire class 1 and let $(f_n)$ be a sequence of continuous functions converging pointwise to $f$. For each $n$, let $g_n$ be a polynomial which is uniformly within $1/n$ of $f_n$ on $[-n,n]$ (by the Weierstrass approximation theorem). Then $(g_n)$ converges pointwise to $f$ as well.