On singular boundary points of complex functions

Abstract

Let f be a complex valued function from the open upper halfplane E of the complex plane. We study the set of all z∈∂E such that there exist two Stoltz angles V 1 , V 2 in E with vertices in z ( i.e. , V i is a closed angle with vertex at z and V i \{ z } ⊂ E , i = 1, 2) such that the function f has different cluster sets with respect to these angles at z . E. P. Dolzhenko showed that this set of singular points is G ∂σ and σ-porous for every f . He posed the question of whether each G ∂σ σ-porous set is a set of such singular points for some f . We answer this question negatively. Namely, we construct a G ∂ porous set, which is a set of such singular points for no function f .