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Fundamenta Mathematicae

1991-1992 | 140 | 1 | 79-85
Tytuł artykułu

Category theorems concerning Z-density continuous functions

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The ℑ-density topology $T_ℑ$ on ℝ is a refinement of the natural topology. It is a category analogue of the density topology [9, 10]. This paper is concerned with ℑ-density continuous functions, i.e., the real functions that are continuous when the ℑ-density} topology is used on the domain and the range. It is shown that the family $C_ℑ$ of ordinary continuous functions f: [0,1]→ℝ which have at least one point of ℑ-density continuity is a first category subset of C([0,1])= {f: [0,1]→ℝ: f is continuous} equipped with the uniform norm. It is also proved that the class $C_ℑℑ$ of ℑ-density continuous functions, equipped with the topology of uniform convergence, is of first category in itself. These results remain true when the ℑ-density topology is replaced by the deep ℑ-density topology.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
79-85
Opis fizyczny
Daty
wydano
1991
otrzymano
1991-02-11
Twórcy
autor
• Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506, U.S.A.
autor
• Department of Mathematics, University of Louisville, Louisville, Kentucky 40292, U.S.A.
Bibliografia
• [1] A. M. Bruckner, Differentiation of Real Functions, Lecture Notes in Math. 659, Springer, 1978.
• [2] K. Ciesielski and L. Larson, Analytic functions are ℑ-density continuous, submitted.
• [3] K. Ciesielski and L. Larson, Baire classification of ℑ-approximately and ℑ-density continuous functions, submitted.
• [4] K. Ciesielski and L. Larson, The space of density continuous functions, Acta Math. Hungar., to appear.
• [5] K. Ciesielski and L. Larson, Various continuities with the density, ℑ-density and ordinary topologies on ℝ, Real Anal. Exchange, to appear.
• [6] K. Ciesielski, L. Larson and K. Ostaszewski, Density continuity versus continuity, Forum Math. 2 (1990), 265-275.
• [7] E. Łazarow, The coarsest topology for I-approximately continuous functions, Comment. Math. Univ. Carolin. 27 (4) (1986), 695-704.
• [8] R. J. O'Malley, Baire*1, Darboux functions, Proc. Amer. Math. Soc. 60 (1976), 187-192.
• [9] W. Poreda, E. Wagner-Bojakowska and W. Wilczyński, A category analogue of the density topology, Fund. Math. 125 (1985), 167-173.
• [10] W. Wilczyński, A category analogue of the density topology, approximate continuity and the approximate derivative, Real Anal. Exchange 10 (1984/85), 241-265.
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Bibliografia
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