On the Baire class of \(n\)-dimensional boundary functions

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 \(\lim_{z\rightarrow x \atop z\in\gamma} f(z) = \varphi(x)\).