The Compactness of the Set of Arc Cluster Sets of an Arbitrary Function

Abstract

It is known that if f is a continuous complex-valued function defined in the open unit disk D, then the set ${\mathfrak {C}_f}(\zeta )\;(\zeta \in \partial D)$ of all arc cluster sets of f at $\zeta$ is compact in a natural topology for all but at most a countable number of points $\zeta \in \partial D$. We show that if f is an arbitrary complex-valued function defined on an arbitrary subset Z of the plane, then ${\mathfrak {C}_f}(p)$ is compact for all but at most a countable number of points $p \in Z \cup \partial Z$.