On characterizing extendable connectivity functions by associated sets

Abstract

We answer two questions asked by Rosen in 1994. We show that the class of extendable connectivity functions from \(I\) into \(I\) (\(I = [0,1]\)) cannot be characterized in terms of associated sets, and we show that one of Jones’ functions obeying \(f(x+y) = f(x) + f(y)\) is an example of an almost continuous function from \(\mathbb{R}\) into \(\mathbb{R}\) which is not the uniform limit of any sequence of extendable connectivity functions.