A holomorphic function with wild boundary behavior

Abstract:

Let $B$ be the open unit ball in ${{\mathbf {C}}^N},N > 1$. It is known that if $f$ is a function holomorphic in $B$, then there are $x \in \partial B$ and an arc $\Lambda$ in $B \cup \left \{ x \right \}$, with $x$ as one endpoint along which $f$ is constant. We prove Theorem. There exist an $r > 0$ and a function $f$ holomorphic in $B$ with the property that, if $x \in \partial B$ and $\Lambda$ is a path with $x$ as one endpoint, such that $\Lambda - \left \{ x \right \}$ is contained in the open ball of radius $r$ which is contained in $B$ and tangent to $\partial B$ at $x$, then $\lim _{z \in \Lambda ,z \to x}f\left ( z \right )$ does not exist.