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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.

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### References:

[1] | BLENDSOE W. W.: Neighbourly functions. Proc. Am. Math. Soc. 3 (1952), 114-115. |

[2] | BRUCKNER A. M., CEDER J. G., PEARSON T. L.: On Darboux functions. Rev. Roum. Math. Pures Appl. 19 (1974), 977-988. · Zbl 0289.26005 |

[3] | GRANDE Z.: Sur les fonctions cliquish. Časopis Pěst. Mat. 110 (1985), 225-236. · Zbl 0579.54009 |

[4] | GRANDE Z., NATKANIEC T.: Lattices generated by r-quasi-continuous functions. Bull. Polish Acad. Sci. Math. 34 (1986), 525-530. · Zbl 0623.26004 |

[5] | 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 |

[6] | GRANDE Z., SOLTYSIK L.: Some remarks on quasi-continuous real functions. Problemy Mat. WSP Bydgoszcz z.10 (1988), 79-87. |

[7] | NATKANIEC T.: Products of quasi-continuous functions. Math. Slovaca 40 (1990), 401-405. · Zbl 0755.26002 |

[8] | NEUBRUNN T.: Quasi-continuity. Real Anal. Exch. 14 (1988-89), 259-306. · Zbl 0679.26003 |

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