Uniformly antisymmetric functions and \(K_ 5\) (Q1912717)

A function \(f: R\to R\) (\(R\) -- the real line) is said to be uniformly antisymmetric provided for every \(x\in R\) there exists \(g(x)> 0\) such that \(|f(x+ h)- f(x- h)|\geq g(x)\) holds for every \(h\), \(0< h< g(x)\). The author deals with the problem of the existence of a uniformly antisymmetric function with a finite range. It is known that there is no uniformly antisymmetric function with two- and three-element range. In the present paper, it is shown that the technique used in the author's previous paper [Real. Anal. Exch. 19, No. 2, 616-619 (1994; Zbl 0806.26002)] in the proof that there is no uniformly antisymmetric function with three-element range cannot be used for the four-element range proof. It is also shown that there exists a uniformly anti-Schwartz function (a function \(f: R\to R\) is said to be uniformly anti-Schwartz if for each \(x\in R\) there exists \(g(x)> 0\) such that \(|f(x- h)+ f(x+ h)- 2f(x)|\geq g(x)\) for every \(0< h< g(x))\).