On the Baire class of $n$-dimensional boundary functions

TL;DR

This paper extends Kaczynski's theorem to boundary functions in n-dimensional space, analyzing the Baire class of such functions and their boundary behavior in higher dimensions.

Contribution

It generalizes a classical theorem to n-dimensional boundary functions, providing new insights into their Baire class and boundary limits.

Findings

Extension of Kaczynski's theorem to n-dimensional boundary functions

Characterization of boundary functions via arcs in higher dimensions

Analysis of the Baire class of boundary functions in n-dimensional space

Abstract

We show an extention of a theorem of Kaczynski to boundary functions in n-dimensional space. Let $H$ denote the upper half-plane, and let $X$ denote its frontier, the $x$-axis. Suppose that $f$ is a function mapping $H$ into some metric space $Y$. If $E$ is any subset of $X$, we will say that a function $\varphi: E \rightarrow Y$ is a boundary function for $f$ if and only if for each $x\in E$ there exists an arc $\gamma$ at $x$ such that $\lim_{z\rightarrow x \atop z\in\gamma} f(z) = \varphi(x)$.