Show that there doesn't exist a sequence of continuous functions $f_n:\Bbb{R}\to\Bbb{R}$ that converges pointwise to $f$ such that
$\forall x\in\Bbb{Q} \space f(x)=1$ and $\forall x\notin\Bbb{Q} \space f(x)=0$.
I know how to prove this using Baire's Category Theorem but I am not allowed to use it in this particular course. Is there proof of this that would not utilize BCT and the likes of it?
A measure-theoretic proof: Suppose that $f_n\to f$ pointwise and that all $f_n$ are continuous. Let $N\in\mathbb{N}$ and $\varepsilon>0$, small. Then since $f_n\to f$ pointwise on $[-N,N]$, by Egorov's theorem, there exists a Lebesgue-measurable subset $E\subset[-N,N]$ such that $m([-N,N]\setminus E)<\varepsilon$ and $f_n\to f$ uniformly on $E$. But then $f$ has to be continuous on $E$, which is impossible, since $f$ is continuous nowhere.