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On the maximum and the minimum of quasi-continuous functions. (English) Zbl 0789.54021
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
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##### References:
  BLENDSOE W. W.: Neighbourly functions. Proc. Am. Math. Soc. 3 (1952), 114-115.  BRUCKNER A. M., CEDER J. G., PEARSON T. L.: On Darboux functions. Rev. Roum. Math. Pures Appl. 19 (1974), 977-988. · Zbl 0289.26005  GRANDE Z.: Sur les fonctions cliquish. Časopis Pěst. Mat. 110 (1985), 225-236. · Zbl 0579.54009 · eudml:21592  GRANDE Z., NATKANIEC T.: Lattices generated by r-quasi-continuous functions. Bull. Polish Acad. Sci. Math. 34 (1986), 525-530. · Zbl 0623.26004  GRANDE Z., NATKANIEC T., STROŃSKA E.: Algebraic structures generated by d-quasi-continuous functions. Bull. Polish Acad. Sci. Math. 35 (1987), 717-723. · Zbl 0655.26007  GRANDE Z., SOLTYSIK L.: Some remarks on quasi-continuous real functions. Problemy Mat. WSP Bydgoszcz z.10 (1988), 79-87.  NATKANIEC T.: Products of quasi-continuous functions. Math. Slovaca 40 (1990), 401-405. · Zbl 0755.26002 · eudml:31805  NEUBRUNN T.: Quasi-continuity. Real Anal. Exch. 14 (1988-89), 259-306. · Zbl 0679.26003
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