 Solution (source code)

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= Solution

The <Urysohn lemma> states that if $A$ and $B$ are disjoint closed subsets of a normal space $X$, there is a continuous $u:X\to[0,1]$ with $u=0$ on $A$ and $u=1$ on $B$.

The <Tietze extension theorem> states that if $A$ is a closed subset of a normal space $X$, then every continuous $f:A\to\mathbb R$ has a continuous extension $F:X\to\mathbb R$. In the bounded form, if $f(A)\subseteq[a,b]$, the extension can be chosen with $F(X)\subseteq[a,b]$.

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