Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 2G (Topics in Analysis)

The closed-set form of the Baire category theorem says: if a nonempty complete metric space is a countable union of closed sets, at least one of those sets has nonempty interior. Equivalently, a complete metric space cannot be a countable union of closed sets with empty interiors. In the application the space is $\mathbb{R}$ with its usual complete metric.

Suppose the continuous functions converged pointwise to the rationality indicator. Each $F_N$ is closed, sincePointwise convergence implies that every point belongs to some $F_N$: its eventual values lie near either zero or one. Thus $\mathbb{R}={\bigcup }_N{F}_N$. By the Baire category theorem, some $F_N$ contains a nonempty open interval $I$. Choose a rational $r\in I$ and an irrational $s\in I$. For sufficiently large $n\ge N$, convergence gives $f_n(r)>2/3$ and $f_n(s)<1/3$. The intermediate value theorem then forces $f_n$ to take the value $1/2$ between them, at a point in $I\subseteq F_N$. This contradicts the defining condition of $F_N$. Therefore the rationality indicator is not a pointwise limit of continuous functions, the rationality indicator is not Baire class one result.