Proposition 5.5.5. Let $X$ be a topological space, $Y$ be a complete Hausdorff uniform space, $A \subset X$ be a dense subset, and $f \in C(A; Y)$ be a continuous function, then the following are equivalent:
There exists a continuous function $F \in C(X; Y)$ such that $F|_{A} = f$.
$f$ is Cauchy continuous.
Proof. By Proposition 5.1.16, $Y$ is regular. Since $Y$ is complete, (2) is equivalent to (2) of Theorem 4.9.2. Therefore the proposition follows from Theorem 4.9.2.$\square$