Generalization of Urysohn’s Lemma, again.

Question

Let $A,B,C$ be closed mutually disjoint sets in a normal space. Can we always find a continuous function $f:\longrightarrow [0,2] $ with $f(A)=0$, $f(B)=1$ and $f(C)=2$? Clearly when we have two sets then this we have the usual Urysohn's Lemma.

Answer 1

Consider $g, h$ be continuous functions to $[0, 1]$ with $g(A \cup B) = 0$ and $g(C) = 1$, and $h(A) = 0$ and $h(B \cup C) = 1$. Then $f = g + h$ is the function you want.