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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)

##### MSC:
 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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