Exercise 23, § 8

Let $X$ be a topological space and let $A$ be a dense subset of $X$ such that the topology of $X$ is $A$-maximal (§ 3, Exercise 11). Let $f$ be a continuous mapping of $A$ into a topological space $Y$, such that at every point of $X - A$ the set of cluster points of $f$ relative to $A$ is not empty (this will be the case, for example, if $X$ is not empty and $Y$ is quasi-compact). Show that $f$ can be extended by continuity to the whole of $X$.