Extensions of vector-valued Baire one functions with preservation of points of continuity

Abstract

We prove an extension theorem (with non-tangential limits) for vector-valued Baire one functions. Moreover, at every point where the function is continuous (or bounded), the continuity (or boundedness) is preserved. More precisely: Let $H$ be a closed subset of a metric space $X$ and let $Z$ be a normed vector space. Let $f: H\to Z$ be a Baire one function. We show that there is a continuous function $g: (X\setminus H) \to Z$ such that, for every $a\in \partial H$, the non-tangential limit of $g$ at a equals $f(a)$ and, moreover, if $f$ is continuous at $a\in H$ (respectively bounded in a neighborhood of $a\in H$) then the extension $F=f\cup g$ is continuous at $a$ (respectively bounded in a neighborhood of $a$). We also prove a result on pointwise approximation of vector-valued Baire one functions by a sequence of locally Lipschitz functions that converges "uniformly" (or, "continuously") at points where the approximated function is continuous. In an accompanying paper (Extensions of vector-valued functions with preservation of derivatives), the main result is applied to extensions of vector-valued functions defined on a closed subset of Euclidean or Banach space with preservation of differentiability, continuity and (pointwise) Lipschitz property.