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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?

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  • $\begingroup$ 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. $\endgroup$ – DanielWainfleet Apr 3 at 5:56
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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.

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