On universally bad Darboux functions (Q1175319)

Let \(D^*=\{f: \mathbb{R}\to\mathbb{R}\); \(\hbox{cl }f^{-1}(y)=\mathbb{R}\) for every \(y\in\mathbb{R}\}\). It is proven that there is an \(f\in D^*\) such that for each continuous and nowhere constant function \(g\) the functions \(f+g\), \(f-g\), \(fg\) and \(f/g\) (if \(0\not\in \hbox{rng} (g))\) do not have the Darboux property.