On a problem concerning universally bad Darboux functions (Q1308848)

A real function \(f\) on \(\mathbb{R}\) is Darboux if \(f(I)\) is connected whenever \(I\) is so. But this property is not stable under addition of continuous \(g\) to \(f\) -- as known since the thirties. Recently it was shown that under, e.g., Martin's axiom there exists a Darboux \(f\) ``universally bad'' from this point of view, i.e., \(f+g\) is not Darboux whenever \(g\) is continuous and nowhere constant. In the paper under review it is shown that the condition ``nowhere constant'' cannot be replaced by the more natural ``nonconstant''. (Another proof of this result is contained in the forthcoming paper \textit{Sum of Darboux and continuous functions} by \textit{J. Steprāns} which moreover shows the ZFC-independence of the existence of an universally bad \(f\)).