On uniformly antisymmetric functions (Q1322630)

A uniformly antisymmetric function is an \(f: R\to R\) (\(R\) -- the real line) such that for every \(x\in R\) there is a \(d(x)> 0\) so that \(0< h< d(x)\) implies \(| f(x+ h)- f(x- h)|\geq d(x)\). The authors investigate uniformly antisymmetric functions by way of the set \(S_ x= \{h>0: f(x- h)= f(x+ h)\}\), \(x\in R\). It is shown, e.g.: \(1^ \circ\) If the continuum hypothesis holds, then there is a function \(f: R\to \omega\) (\(\omega\) -- the set of natural numbers) such that for every \(x\in R\;\;S_ x\) has at most one element; \(2^ \circ\) There is a function \(f: Q\to \{0,1,2,3\}\) (\(Q\) -- the set of rationals) such that for every \(x\in Q\;\;S_ x\) is finite.