On porosity and exceptional sets (Q1078693)

Let f be a real valued function defined in the upper half plane \(\{\) (x,y): \(y>0\}\). Let E(f) denote the set of all points x on the real line where f has unequal angular cluster sets on any two Stolz angles with vertexes at the point x. Theorem. If E is a \(G_{\delta \sigma}\), sigma- globally porous set on the real line, then there is a continuous function f from the upper half plane to [0,1], such that \(E=E(f)\). This is a partial answer to a question of \textit{H. Yoshida} [Nagoya Math. J. 46, 111-120 (1972; Zbl 0211.389)]. The paper does not contain the proof of the Theorem. The notion of the sigma-globally porous set was introduced by \textit{P. D. Humke} and the author [Real. Anal. Exch. 8, 262-271 (1983; Zbl 0543.28001)].