Uniformly Antisymmetric Functions With Bounded Range

Abstract

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 lim n→∞ (f (x + s n ) − f (x − s n )) equals 0 for no sequence {s n } 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 n 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.

On nowhere weakly symmetric functions and functions with two-element range

“…In [2] Ciesielski proved that there is no nowhere weakly symmetrically continuous function with three-element range and in [3] he showed that the technique of [2] cannot be used to prove that there is no nowhere weakly symmetrically continuous function with four-element range. It is still an open question (see [7,Problem 1(a)], [6, Problem 2(a)], or [12]) whether there exists a nowhere weakly symmetrically continuous function with finite range, though Ciesielski and Shelah [8] constructed a nowhere weakly symmetrically continuous function with bounded countable range.…”

Points of Weak Symmetric Continuity

“…We do not know if the same is true for 4-valued functions. Recently Ciesielski and Shelah [4] showed that there is an everywhere weakly symmetrically discontinuous function with bounded range.…”