On the maximum and the minimum of quasi-continuous functions. (English) Zbl 0789.54021

Natkaniec, Tomasz

Math. Slovaca 42, No. 1, 103-110 (1992).

The paper contains three theorems. In the first is generalized some result of Z. Grande and L. Sołtysik from [Zesz. Nauk. Wyższ. Szk. Pedagog. Bydgoszczy, Probl. Math. 10, 79-86 (1990; Zbl 0705.26008)]. They proved that if a function \(f:X \to \mathbb{R}\) is not upper (lower) semicontinuous, then there exists a quasi-continuous function \(g:X \to \mathbb{R}\) such that \(\max (f,g)\) \((\min (f,g))\) is not quasi- continuous. Now it is proved that for every non-continuous function \(f:X \to \mathbb{R}\) there exist two quasi-continuous functions \(g,h:X \to \mathbb{R}\) for which \(\max(f,g)\) and \(\min (f,h)\) are not quasi-continuous.
In the second part the lattice generated by the family of all quasi- continuous functions \(f:X \to \mathbb{R}\) (for some class of topological spaces \(X)\) is described. This generalizes a result from [Z. Grande and the author, Bull. Pol. Acad. Sci., Math. 34, 525-530 (1986; Zbl 0623.26004)] (for \(X=\mathbb{R})\). Notice that in the proof in the just cited paper the completeness of \(\mathbb{R}\) plays the key role. Now this assumption is not necessary.
The last proposition characterizes \(\max(g_ 0,g_ 1)\) for quasi- continuous functions \(g_ 0,g_ 1:\mathbb{R} \to \mathbb{R}\). This is a complement of results from [Z. Grande, Čas. Pěstování Mat. 110, 225-236 (1985; Zbl 0579.54009)] and [the author, Math. Slovaca 40, No. 4, 401-405 (1990; Zbl 0755.26002)] where functions which can be expressed as sum and product of a finite number of quasi-continuous functions are characterized.

MSC:

54C30	Real-valued functions in general topology
54C08	Weak and generalized continuity
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Keywords:

quasi-continuity; Darboux property; lattice of quasi-continuous functions; quasi-continuous function

Citations:

Zbl 0705.26008; Zbl 0623.26004; Zbl 0579.54009; Zbl 0755.26002

References:

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