Universally bad Darboux functions in the class of additive functions

Abstract

The main result: For every family \(\mathcal{G}\) of additive functions with \(\text{card }{\mathcal{G}}=2^\omega\) if the covering of the family of all level sets of functions from \(\mathcal{G}\) is equal to \(2^\omega\), then there exists an additive Darboux function \(f\) such that \(f+g\) is Darboux for no \(g\in\mathcal{G}\).