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

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 $\phi :E\to Y$ is a boundary function for $f$ if and only if for each $x\in E$ there exists an arc $\gamma$ at $x$ such that ${\lim }_{\genfrac{}{}{0ex}{}{z\to x}{z\in \gamma }}f(z)=\phi (x)$.

3 pages, 0 figures