$D \subset \mathbb R$ and we have two functions $U: D \to \mathbb R$ and $L: D \to \mathbb R$, with the given property that $\forall x \in D: U(x) > L(x)$.
Because $U(x) \neq L(x)$ there are infinitely many rational numbers between them. I want to choose one, and call it $q(x)$. It does not matter which one.
Infact I want to define a function $q:D \to \mathbb Q$ such that $L(x) < q(x) < U(x)$. Is it possible to define such a function without explicitly saying what $q(x)$ is?
Note: $D$ does not have to be countable.
You can take the shortcut via the Axiom of Choice, which handles precisely this case (even for domains $D$ much larger than $\Bbb R$).
Or you can even define $q(x)$ constructively: Given $x$, let $$n=\min\left\{\,k\in\Bbb N\biggm| k>\frac1{U(x)-L(x)}\,\right\}$$ and $$m = \min (\Bbb Z\cap (nL(x),nU(x))).$$ Then $$L(x)<\frac mn<U(x).$$