Darboux Like Functions that are Characterizable by Images, Preimages and Associated Sets

Abstract

For \(\mathcal{A},\mathcal{B}\subset\mathcal{P}(\mathbb{R})\) let \(\mathcal{C}_{\mathcal{A},\mathcal{B}}=\{ f\in\mathbb{R}^\mathbb{R}\colon(\forall A\in\mathcal{A})\,(f(A)\in\mathcal{B})\}\) and \(\mathcal{C}_{\mathcal{A},\mathcal{B}}^{-1}=\{ f\in\mathbb{R}^\mathbb{R}\colon(\forall B\in\mathcal{B})\,(f^{-1}(B)\in\mathcal{A})\}\). A family \(\mathcal{F}\) of real functions is characterizable by images (preimages) of sets if \(\mathcal{F}=\mathcal{C}_{\mathcal{A},\mathcal{B}}\) (\(\mathcal{F}=\mathcal{C}_{\mathcal{A},\mathcal{B}}^{-1}\), respectively) for some \(\mathcal{A},\mathcal{B}\subset\mathcal{P}(\mathbb{R})\). We study which of the classes of Darboux like functions can be characterized in this way. Moreover, we prove that the class of all Sierpiński-Zygmund functions can be characterized by neither images nor preimages of sets.