Uniformly Antisymmetric Functions

Uniformly Antisymmetric Functions With Bounded Range

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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].)

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“…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].)…”

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

“…In [7] Ciesielski and Larson constructed a nowhere weakly symmetrically continuous function f : R → N for which each set S x is finite. (See also [5,Cor.…”

“…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.…”

“…It should be pointed out here that the equivalences (i)⇔(ii)⇔(iii) are known and were proved in [7]. (See also [5,Thm.…”