Uniformly Antisymmetric Functions with Bounded Range

Abstract

The goal of this note is to construct a uniformly antisymmetric function $f\colon\mathbb{R}\to\mathbb{R}$ 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 $f\colon\mathbb{R}\to\mathbb{R}$ with finite range remains open.