The continuity of symmetric and smooth functions

Abstract

which are non-measurable, fail to possess the Baire property and assume every real value uncountably many times on every perfect set [4]. However, symmetric functions which are measurable are known to be quite nice in that they are of Baire class one and are continuous almost everywhere [5]. As might be expected, measurable smooth functions are even better behaved in that they belong to the class Baire* one [7] and have only a countable number of discontinuities [6]. In this work the set of points on which symmetric and smooth functions are continuous is studied. First, it is shown that an arbitrary function with the Baire property which is symmetric on a residual set of points is also continuous on a residual set. In then follows easily that any symmetric function with the Baire property is in Baire class one. Second, the set of points at which a measurable smooth function is discontinuous is characterized as a separated set in the sense of Hausdorff [3]. First, some notation must be introduced. I f A c R , A will denote the closure of A, A" will denote the set of limit points of A, and A c will denote the complement of A. The distance between a point x and the set A will be denoted by d(x, A). The set A = R is separated if A has no subset which is dense in itself. All functions are finite-valued with domains contained in R. If f is a function, then the oscillation of f at x is written o)f(x). The set of points at which f is continuous is written C(f) and the set of points at which f is discontinuous is written D(f). A function f : R R has the Baire property if there is a set A residual in R such that the restriction of f to A, f IA, is continuous. If f is symmetric at x and ~>0, then 6(x, ~) denotes a positive number such that ]A2f(x,h)]<~ for O<h<5(x, e).