Extensions of continuous functions from dense subspaces

Abstract:

Let $X$ and $Y$ be topological spaces, let $S$ be a dense subspace of $X$, and let $f:S \to Y$ be continuous. When $Y$ is the real line ${\mathbf {R}}$, the Lebesgue sets of $f$ are used to provide necessary and sufficient conditions in order that the (bounded) function $f$ have a continuous extension over $X$. These conditions yield the theorem of Taǐmanov (resp. of Engelking and of Blefko and Mrówka) which characterizes extendibility of $f$ for $Y$ compact (resp. realcompact). In addition, an extension theorem of Blefko and Mrówka is sharpened for the case in which $X$ is first countable and $Y$ is a closed subspace of ${\mathbf {R}}$.