On some Modification of Świtąkowski Property

Ivanova, Gertruda; Wagner-Bojakowska, Elzbieta

doi:10.2478/tmmp-2014-0009

Abstract

We introduce some families of functions ƒ : ℝ → ℝ modifying the Darboux property analogously as it was done by [Maliszewski, A.: On the limits of strong Świątkowski functions, Zeszyty Nauk. Politech. Łódź. Mat. 27 (1995), 87-93], replacing continuity with A-continuity, i.e., the continuity with respect to some family A of subsets in the domain. We prove that if A has (*)-property then the family DA of functions having A-Darboux property is contained and dense in the family DQ of Darboux quasi-continuous functions.

References

[1] BRUCKNER, A. M.: Differentiation of Real Functions, in: Lecture Notes in Math., Vol. 659, Springer, Berlin, 1978.10.1007/BFb0069821
Search in Google Scholar
[2] BRUCKNER, A. M.-CEDER, J. G.-WEISS, M. L.: Uniform limits of Darboux functions, Colloq. Math. 15 (1966), 65-77.10.4064/cm-15-1-65-77
Search in Google Scholar
[3] CIESIELSKI, K.-LARSON, L.-OSTASZEWSKI, K.: I-density continuous functions, Mem. Amer. Math. Soc. 515 (1994), 133 p.10.1090/memo/0515
Search in Google Scholar
[4] DARBOUX, G.: Memoire sur les fonctions discontinues, Ann. Sci. Scuola Norm. Sup. 4 (1875), 57-112.
Search in Google Scholar
[5] GRANDE, Z.: On a subclass of the family of Darboux functions, Colloq. Math. 117 (2009), 95-104.10.4064/cm117-1-6
Search in Google Scholar
[6] IVANOVA, G.: Remarks on some modification of the Darboux property, Bull. Soc. Sci. Lettres Ł´odź Sér. Rech. Déform. 72 (2014), 91-100.
Search in Google Scholar
[7] IVANOVA, G.-WAGNER-BOJAKOWSKA, E.: On some modification of Darboux property, Math. Slovaca (to appear).
Search in Google Scholar
[8] HASHIMOTO, H.: On the *-topology and its application, Fund. Math. 91 (1976), 5-10.10.4064/fm-91-1-5-10
Search in Google Scholar
[9] KURATOWSKI, A.-MOSTOWSKI,A.: Teoria MnogoŚciWraz zeWst¸epem do Opisowej Teorii MnogoŚci. PWN, Warszawa, 1978.
Search in Google Scholar
[10] ŁAZAROW, E.-JOHNSON, R. A.-WILCZYŃSKI, W.: Topologies related to sets having the Baire property, Demonstratio Math. 21 (1989), 179-191.10.2478/dema-1989-0117
Search in Google Scholar
[11] LEVINE, N.: Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41.10.1080/00029890.1963.11990039
Search in Google Scholar
[12] LINDENBAUM, A.: Sur quelques propriétés des fonctions de variable réelle, Ann. Soc. Math. Polon. 6 (1927), 129-130.
Search in Google Scholar
[13] MALISZEWSKI, A.: Darboux Property and Quasi-Continuity. A uniform approach. WSP, SŁupsk, 1996.
Search in Google Scholar
[14] MALISZEWSKI, A.: On the limits of strong Świ¸atkowski functions, Zeszyty Nauk. Politech. dz. Mat. 27 (1995), 87-93.
Search in Google Scholar
[15] MALISZEWSKI, A.: Sums and products of quasi-continuous functions, Real Anal. Exchange 21 (1995-96), 320-329.10.2307/44153922
Search in Google Scholar
[16] MAŃK, T.-ŚWIĄTKOWSKI, T.: On some class of functions with Darboux’s characteristic, Zeszyty Nauk. Politech. dz. Mat. 11 (301) (1977), 5-10.
Search in Google Scholar
[17] NATKANIEC, T.: On quasi-continuous functions having Darboux property, Math. Pannon. 3 (1992), 81-96.
Search in Google Scholar
[18] NEUBRUNNOVÀ, A.: On certain generalizations of the notion of continuity, Matematick ý Časopis 23 (1973), 374-380.
Search in Google Scholar
[19] POREDA, W.-WAGNER-BOJAKOWSKA, E.-WILCZYŃSKI, W.: A category analogue of the density topology, Fund. Math. 125 (1985), 167-173.10.4064/fm-125-2-167-173
Search in Google Scholar
[20] POREDA, W.-WAGNER-BOJAKOWSKA, E.-WILCZYŃSKI, W.: Remarks on I-density and I-approximately continuous functions, Comm. Math. Univ. Carolinae 26 (1985), 553-563.
Search in Google Scholar
[21] SIERPIŃSKI, W.: Sur une propriété de fonctions réelles quelconques définies dans les espaces métriques, Matematiche (Catania) 8 (1953), 73-78.
Search in Google Scholar
[22] WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273-286.10.14321/realanalexch.32.1.0273
Search in Google Scholar
[23] WILCZYŃSKI, W.: A generalization of the density topology, Real Anal. Exchange 8 (1982-83), 16-20.10.2307/44151572
Search in Google Scholar
[24] WILCZYŃSKI, W.: A category analogue of the density topology, approximate continuity and the approximate derivative, Real Anal. Exchange 10 (1984-85), 241-265. 10.2307/44153572
Search in Google Scholar

On some Modification of Świtąkowski Property

Abstract

Paradigm

My account