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,\Deltax a triangle completely lying in H except for its vertex at x, and C(f,x,\Deltax) the cluster set of f at x obtained along \Deltax. 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,\Deltax) = 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,\Deltax) = {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 Lj 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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