Some consequences of the Freiling-Humke result on the density property (Q1322634)

In the paper are shown some of the consequences of results by \textit{C. Freiling} and \textit{P. D. Humke} [Real Anal. Exch. 17, No. 1, 272-281 (1992; Zbl 0756.28001)] concerning the density property of certain sets. A function \(f: \mathbb{R}\to\mathbb{R}\) is said to be uniformly positively approached from below if there is an \(\alpha> 0\) such that \(\{x: f(x)> y\}\) has upper density \(>\alpha\) at all its points for every \(y\in \mathbb{R}\). The main result of the paper is the following one: If \(f\) is approximately upper semicontinuous on \([0,1]\) and uniformly positively approached from below, then there is \(x_ 0\in [0,1]\) at which \(f\) has an approximate maximum. That is, at \(x_ 0\) the set \(\{x: f(x)> f(x_ 0)\}\) has density 0. The statement improves some known results.