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On some subclasses of Darboux functions. (English) Zbl 0749.26003
Let \(\mathcal X\) be a class of real functions defined in some topological spaces. The authors denote the maximal additive (multiplicative, lattice- like) class for \(\mathcal X\) by \({\mathcal M}_ a({\mathcal X})\) \(({\mathcal M}_ m({\mathcal X})\), \({\mathcal M}_ l({\mathcal X}))\). \[ {\mathcal M}_{\min}({\mathcal X})=\{f\in{\mathcal X};\text{ if } g\in{\mathcal X},\text{ then }\min(f,g)\in{\mathcal X}\} \] and \[ {\mathcal M}_{\max}({\mathcal X})=\{f\in{\mathcal X};\text{ if } g\in{\mathcal X},\text{ then } \max(f,g)\in{\mathcal X}\}. \] In this paper the authors show that: \[ \begin{alignedat}{2} &{\mathcal M}_ a({\mathcal A})={\mathcal M}_ l({\mathcal A})={\mathcal C}\text{ and } {\mathcal M}_ m({\mathcal A})={\mathcal M}& &(\text{Theorem 3.1});\\ &{\mathcal M}_ a({\mathcal C}on)={\mathcal M}_ l({\mathcal C}on)={\mathcal C}, {\mathcal M}_ m({\mathcal C}on)={\mathcal M}& &(\text{Theorem 4.1});\\ &{\mathcal M}_ a({\mathcal F})={\mathcal M}_ l({\mathcal F})={\mathcal C}, {\mathcal M}_ m({\mathcal F})={\mathcal M}, {\mathcal M}_{\min}({\mathcal F})={\mathcal D}\cap lsc, {\mathcal M}_{\max}({\mathcal F})={\mathcal D}\cap usc & &(\text{Theorem 5.1});\end{alignedat} \] where \({\mathcal C}on\), \(\mathcal C\), \(\mathcal A\), \(\mathcal D\), \(\mathcal F\), \(lsc\), \(usc\), and \(\mathcal M\) denote the classes of connected functions, continuous functions, almost continuous functions, Darboux functions, functionally connected functions, lower semicontinuous functions, upper semicontinuous functions and the class of Darboux functions \(f\) with the following property: if \(x_ 0\) is a right-hand (left-hand) point of discontinuity of \(f\), then \(f(x_ 0)=0\) and there is a sequence \((x_ n)\) converging to \(x_ 0\) such that \(x_ n>x_ 0\) \((x_ n<x_ 0)\) and \(f(x_ n)=0\).
Reviewer: R.Pawlak (Łódź)

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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