Uniformly antisymmetric function with bounded range

The goal of this note is to construct a uniformly antisymmetric function f : R → R with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson [6]. (See also the list of problems in Thomson [9] and Problem 2(b) from Ciesielski’s survey [5].) A problem of existence of uniformly antisymmetric function f : R → R with finite range remains open. A function f : R → R is said to be uniformly antisymmetric [6] (or nowhere weakly symmetrically continuous [9]) provided for every x ∈ R the limit limn→∞(f(x+ sn)− f(x− sn)) equals 0 for no sequence {sn}n<ω converging to 0. Uniformly antisymmetric functions have been studied by Kostyrko [7], Ciesielski and Larson [6], Komjáth and Shelah [8], and Ciesielski [1, 2]. (A connection of some of these results to the paradoxical decompositions of the Euclidean space R is described in Ciesielski [3].) In particular in [6] the authors constructed a uniformly antisymmetric function f : R → N and noticed that the existence of a uniformly antisymmetric function cannot be proved without an essential use of the axiom of choice.