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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 $[0, 1]$?

Student
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    See here: http://math.stackexchange.com/questions/77307/existence-of-a-sequence-of-continuous-functions-pointwise-convergence – Joey Zou Nov 29 '15 at 02:25
  • The general idea that should come to mind is this: suppose that ${f_n}$ is a sequence of functions converging pointwise to a function $f$. If each $f_n$ has "Property X" does that imply anything at all about what property that $f$ should have? That way you can spare yourself the grief perhaps of having to construct an example of a sequence of Property X functions converging to some proposed example. – B. S. Thomson Nov 29 '15 at 02:58

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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.

BrianO
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    If $f$ is a pointwise limit of continuous real functions then $f^{-1}{y}={x :f(x)=y}$ is a $G_{\delta}$ set for every $y$. A corollary to the Baire Category Theorem is that a dense $G_{\delta}$ set of reals is uncountable. – DanielWainfleet Nov 29 '15 at 04:53