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On singular boundary points of complex functions. (English) Zbl 0903.30024

Let \(f\) be a complex valued function from the open upper half-plane \(E\) of the complex plane. We study the set of all \(z\in\partial E\) 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 \(V_i\setminus\{z\}\subset E\), \(i= 1,2\)) 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 \(G_{\delta\sigma}\) and \(\sigma\)-porous for every \(f\). He posed the question of whether each \(G_{\delta\sigma}\) \(\sigma\)-porous set is a set of such singular points for some \(f\). We answer this question negatively. Namely, we construct a \(G_\delta\) porous set, which is a set of such singular points for no function \(f\).
Reviewer: M.Zelený (Praha)

MSC:

30D40 Cluster sets, prime ends, boundary behavior
26B99 Functions of several variables
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References:

[1] Zajíček, Real Analysis Exchange 13 pp 314– (1987)
[2] Vessey, Real Analysis Exchange 9 pp 336– (1983)
[3] Dolzhenko, Izv. Akad. Nauk. SSSR Ser. Mat. 31 pp 3– (1967)
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