M.: Points of weak symmetry

We show that every set of reals is a set of points of weakly sym-metry for some function f:R → N. 1 Preface Let us begin with a historical background. Probably the weakest notion of continuity is the following definition: Definition 1.1. A function f: R → R is weakly (or peripherally) continuous at x if limn→ ∞ f(xn) = f(x) for some sequence xn → x. The following theorem characterizes the sets of points of weak continuity: Theorem 1.2 (Chapter 2 of [T] and Theorem 4 of [MS]). Any function has only countably many points of weak discontinuity and any countable set is the set of points of weak discontinuity for some function. 2 Notation and Definitions Basic notion for our investigations is the following definition (see for example [CL], [CMN] or [MS]): Definition 2.1. A function f:R → R is weakly symmetrically continuous at a point x if there exists a sequence 〈hn 〉 of positive numbers converging to 0 such that lim n→∞ f(x+ hn) − f(x − hn) = 0. 1Both authors partially supported by grant BW/5100-5-0201-