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

TL;DR

This paper develops an extension theorem for vector-valued Baire one functions, ensuring the preservation of continuity and boundedness at points of the original function, and introduces approximation methods by Lipschitz functions.

Contribution

It introduces a novel extension method for vector-valued Baire one functions that preserves continuity and boundedness, with applications to derivative-preserving extensions.

Findings

Extension theorem with non-tangential limits for Baire one functions

Preservation of continuity and boundedness at points of the original function

Approximation of Baire one functions by locally Lipschitz functions

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.