The result stated in the
title is proved. No restriction other than the obvious requirement that the
cluster sets be taken at a nonisolated boundary point is imposed on the
domain where the mapping is defined. The result is then generalized by
allowing for certain exceptional sets on the boundary. More refined versions are
established in the special case where the domain is the open unit disk. These
include the statement that one-sided cluster sets coincide with one-sided radial
cluster sets. Again, certain exceptional sets on the boundary are allowed for.
Consequences are presented in which the existence of limits along sets on the
boundary implies limits inside the domain. Finally, generalizations to the class of
homeomorphisms satisfying the Carathéodory Prime End Theorem are
indicated.