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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 113.05503

**Autor: ** Erdös, Pál; Piranian, G.

**Title: ** Restricted cluster sets (In English)

**Source: ** Math. Nachr. 22, 155-158 (1960).

**Review: ** Let f be a complex-valued function in the upper half plane H,x a point on the real axis, C(f,x) the (ordinary) cluster set of f at x,\Delta_{x} a triangle completely lying in H except for its vertex at x, and C(f,x,\Delta_{x}) the cluster set of f at x obtained along \Delta_{x}. Independently of *E.F.Collingwood* [Proc. Natl. Acad. Sci. USA 46, 1236-1242 (1960; Zbl 142.04401)], the authors obtain the same result that there exists a residual set of x for which \cap_{\Deltax} C(f,x,\Delta_{x}) = C(f,x). Secondly, given a set E of first category on the real axis, the existence of f in H having the following properties is shown: \cup_{\Deltax} C(f,x,\Delta_{x}) = **{**0**}** for each x in E and C(f,x) is identical to the extended plane for each x. Finally the authors extend a result of *F.Bagemihl, G.Piranian* and *G.S.Young* [Bul. Inst. Politehn. Iasi, n. Ser. 5, 29-34 (1959; Zbl 144.33203)] according to which there exists a function in H with the property that each x is the endpoint of three segments L_{j} such that the cluster sets along them have no point in common. A correction to the proof of Theorem 2 is given in MR 23.A1041 (1962).

**Reviewer: ** M.Ohtsuka

**Classif.: ** * 28-99 Measure and integration

**Index Words: ** complex functions

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