An abstract formulation of some theorems on cluster sets

Abstract

There are a number of results in the literature which affirm the symmetric structure of functions in terms of Cluster Sets. For example, W. H. Young [5] showed that an arbitrary function f: R-*R has a remarkable symmetry property which may be stated in terms of Cluster Sets. A real number y is a right limit of f at xER if there exists a sequence xn>x, n= 1, 2, * , such that lim xn,=x and limf(xn) =y. A left limit of f at xER is defined similarly. These sets we designate by C+(f, x) and C(f, x), respectively. The result of Young asserts that C+(f, x) = C(f, x) at every xER-A where A is a countable set. Young obtained an analogous result for functions of two variables. By a sector at the origin, we mean either of the closed regions formed by two half-lines emanating from the origin. If ois a sector at the origin, then we designate by o-, the image of ounder the translation taking the origin into p. Letf: R2 >*R be arbitrary and let p CR2. The Cluster Set, C(f, p), of f at p is the set of all real numbers w for which there exists a sequence { pn } CR2 such that lim Pn = p and lim f(pn) = w. If ois a sector at the origin, then The Sectorial Cluster Set, C(f, p, o-,), is defined in the obvious manner. The result Young obtained in this case is that