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.
Citation
Krzysztof Ciesielski. "Darboux Like Functions that are Characterizable by Images, Preimages and Associated Sets." Real Anal. Exchange 23 (2) 441 - 458, 1997/1998.
Information