TITLE: Exists continuous $f_n: [0,1] \to \mathbb{R}$ that converges pointwise, as $n \to \infty$, to $\chi_\mathbb{Q}$?
QUESTION [5 upvotes]: Does there exist a sequence of continuous $f_n: [0, 1] \to \mathbb{R}$ that converges pointwise, as $n \to \infty$, to $\chi_\mathbb{Q}$, the characteristic function of the rationals in $[0, 1]$?

REPLY [2 votes]: There cannot be such a sequence.
$\chi_{\Bbb Q}$ is a double limit of continuous functions, cf. the Dirichlet function. Thus, it's a Baire class 2 function. As the article states, it can't be a Baire class 1 function (single pointwise limit of continuous functions) because such functions have a meager set of discontinuities, unlike $\chi_{\Bbb Q}$. For a proof, see the links and reference in the stackexchange Q&A which Joey Zou cites in his comment.