Urysohn's Lemma extended to general case

Question

I have proved the case for existence of a continuous function $f:X \rightarrow \mathbb{R}$ s.t $f=1$ when restricted to A and $f=0$ when restricted to B. The function I used was $\{r \in \mathbb{Q} : x \in U_r\}$. How can I extend this so that if $A,B$ disjoint subsets of a normal topological space X, then $\forall alb \in \mathbb{R}, \exists f:X \rightarrow \mathbb{R}$ such that $f = a$ on $A$ and $f=b$ on $B$

Answer 1

It's not clear to me what your function is, (nor that it's continuous if it only takes values in $\mathbb Q$?), but given the existence of a function $f\colon X \to \mathbb R$ with $f(x) = 1$ when $x \in A$ and $f(x) = 0$ when $x \in B$, you can replace $1$ and $0$ by $a,b \in \mathbb R$ by postcomposing with the function $g \colon [0,1] \to [a,b]$ defined as $g(t) = (b-a)t + a$.