Uniformly antisymmetric functions (Q1322631)

A function \(f: R\to R\) (\(R\) -- the real line) is said to be uniformly antisymmetric if for every \(x\in R\) there is a \(d_ x> 0\) such that \(| f(x+ h)- f(x- h)|\geq d_ x\), whenever \(0< h< d_ x\). The main result of the paper is a proof of the existence of a uniformly antisymmetric function. Further properties of such functions are also discussed (e.g., if \(f\) is uniformly antisymmetric, then \(f\) is neither measurable nor does it have the Baire property).