Restricted cluster sets

Let f be a complex-valued function defined in the upper half-plane H. The cluuster set C(f, x) at a point x on the real axis is defined as the set of all values w (including possibly w = m) for which there exists a sequence {xJ = (x~ + i Y,J such that x, + x, yn 4 0, and f (2,) -+ WI, We use the term restricted cluster set generically: the word “restricted” indicates either that the sequence {z+J occurring in the definition above is subject to special conditions (for exampk, that it lie on a line segment), or that the cluster set C(f, x, *) is defined in terms of unions or intersections of certain “primitive” cluster sets.