Existence of a sequence of positive continuous functions on $\mathbb R$ which is unbounded precisely on $\mathbb Q$

Is it possible to find a sequence of positive continuous functions $g_n$ on the real numbers such that $( g_n(x) )$ is unbounded if and only if $x \in \mathbb{Q}$ ?

Solution 1:

No, that is not possible. If $(g_\alpha)_{\alpha\in A}$ is a family of continuous real-valued functions defined on $\mathbb{R}$, then $g\colon \mathbb{R} \to (-\infty,+\infty]$ defined by

$$g(x) = \sup \{ g_\alpha(x) : \alpha \in A\}$$

is a lower semicontinuous function. Hence, for all $c\in (-\infty,+\infty]$, the set

$$F(c) := \{ x\in \mathbb{R} : g(x) \leqslant c\}$$

is closed. If $\{ g_\alpha(x) : \alpha \in A\}$ is unbounded from above, i.e. $g(x) = +\infty$, for all $x\in \mathbb{Q}$, then $F(c)$ has empty interior for every $c\in \mathbb{R}$, and thus by Baire's theorem the set

$$B = \bigcup_{n\in\mathbb{N}} F(n) = \{ x\in \mathbb{R} : g(x) < +\infty\}$$

is meagre. But $\mathbb{R}\setminus \mathbb{Q}$ is a non-meagre subset of $\mathbb{R}$.