# What is the set of pointwise limits of polynomials?

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?

• As an adjunct to the accepted answer: If $f:\Bbb R\to \Bbb R$ is Baire-1 then $f^{-1}\{r\}$ is a $G_{\delta}$ set for any $r\in \Bbb R,$ and if $U$ is open in $\Bbb R$ (indeed, if $U$ is any countable union of real intervals) then $f^{-1}U$ is an$F_{\sigma \delta}$ set.... In particular, the characteristic function $\chi_{\Bbb Q}$ of the rationals is $not$ Baire-1 because $(\chi_{\Bbb Q})^{-1}\{1\}=\Bbb Q$ is not a $G_{\delta}$ set. – DanielWainfleet Apr 3 at 5:56

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.