Methods of comparison of families of real functions in porosity terms
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Stanisław Kowalczyk
und
Małgorzata Turowska
Abstract
We consider some families of real functions endowed with the metric of uniform convergence. In the main results of our work we present two methods of comparison of families of real functions in porosity terms. The first method is very general and may be applied to any family of real functions. The second one is more convenient but can be used only in the case of path continuous functions. We apply the obtained results to compare in terms of porosity the following families of functions: continuous, absolutely continuous, Baire one, Darboux, also functions of bounded variation and porouscontinuous, ρ-upper continuous, ρ-lower continuous functions.
References
[1] M. Bienias, S. Gła̧b and W. Wilczyński, Cardinality of sets of ρ-upper and ρ-lower continuous functions, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 64 (2014), no. 2, 71–80. Suche in Google Scholar
[2] J. Borsík and J. Holos, Some properties of porouscontinuous functions, Math. Slovaca 64 (2014), no. 3, 741–750. 10.2478/s12175-014-0237-3Suche in Google Scholar
[3] A. M. Bruckner, Differentiation of Real Functions, Lecture Notes in Math. 659, Springer, Berlin, 1978. 10.1007/BFb0069821Suche in Google Scholar
[4] A. M. Bruckner, R. J. O’Malley and B. S. Thomson, Path derivatives: A unified view of certain generalized derivatives, Trans. Amer. Math. Soc. 283 (1984), no. 1, 97–125. 10.1090/conm/042/807973Suche in Google Scholar
[5] M. Filipczak, G. Ivanova and J. Wódka, Comparison of some families of real functions in porosity terms, Math. Slovaca 67 (2017), no. 5, 1155–1164. 10.1515/ms-2017-0039Suche in Google Scholar
[6] T. Filipczak, On some abstract density topologies, Real Anal. Exchange 14 (1988/89), no. 1, 140–166. 10.2307/44153634Suche in Google Scholar
[7] G. Ivanova, A. Karasińska and E. Wagner-Bojakowska, Comparison of some subfamilies of functions having the Baire property, Tatra Mt. Math. Publ. 65 (2016), 151–159. 10.1515/tmmp-2016-0013Suche in Google Scholar
[8] G. Ivanova and E. Wagner-Bojakowska, On some subclasses of the family of Darboux Baire 1 functions, Opuscula Math. 34 (2014), no. 4, 777–788. 10.7494/OpMath.2014.34.4.777Suche in Google Scholar
[9] S. Kowalczyk and K. Nowakowska, A note on ϱ-upper continuous functions, Tatra Mt. Math. Publ. 44 (2009), 153–158. 10.2478/v10127-009-0055-0Suche in Google Scholar
[10]
S. Kowalczyk and K. Nowakowska,
A note on the
[11] S. Kowalczyk and M. Turowska, On structural properties of porouscontinuous functions, Math. Slovaca 67 (2017), no. 5, 1239–1250. 10.1515/ms-2017-0045Suche in Google Scholar
[12] L. Zajíček, Porosity and σ-porosity, Real Anal. Exchange 13 (1987/88), no. 2, 314–350. 10.2307/44151885Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- On absolutely Baire nonmeasurable functions
- Existence of special rainbow triangles in weak geometries
- Images of Bernstein sets via continuous functions
- Square-integrable representations and multipliers
- Semigroups of pathological sets
- Points of density and ideals of subsets of ℕ
- On Kharazishvili type measures in infinite-dimensional Polish vector spaces
- Families of symmetric Cantor sets from the category and measure viewpoints
- On the Gitik–Shelah theorem
- On equidistant lines of given line configurations
- On convergence of sequences of functions possessing closed graphs
- On combinatorial and set-theoretical aspects of some finite and infinite point sets
- On Kuratowski partitions in the Marczewski and Laver structures and Ellentuck topology
- Automorphism groups of mono-unary algebras and CH
- On topologies related to the extension of the Lebesgue measure
- An abstract formulation of a theorem of Sierpiński on the nonmeasurable sum of two measure zero sets
- Local affine selections of convex set-valued functions
- On orbits without the Baire property
- Mann iteration process for monotone nonexpansive mappings with a graph
- Mixing coded systems
- Methods of comparison of families of real functions in porosity terms
Artikel in diesem Heft
- Frontmatter
- On absolutely Baire nonmeasurable functions
- Existence of special rainbow triangles in weak geometries
- Images of Bernstein sets via continuous functions
- Square-integrable representations and multipliers
- Semigroups of pathological sets
- Points of density and ideals of subsets of ℕ
- On Kharazishvili type measures in infinite-dimensional Polish vector spaces
- Families of symmetric Cantor sets from the category and measure viewpoints
- On the Gitik–Shelah theorem
- On equidistant lines of given line configurations
- On convergence of sequences of functions possessing closed graphs
- On combinatorial and set-theoretical aspects of some finite and infinite point sets
- On Kuratowski partitions in the Marczewski and Laver structures and Ellentuck topology
- Automorphism groups of mono-unary algebras and CH
- On topologies related to the extension of the Lebesgue measure
- An abstract formulation of a theorem of Sierpiński on the nonmeasurable sum of two measure zero sets
- Local affine selections of convex set-valued functions
- On orbits without the Baire property
- Mann iteration process for monotone nonexpansive mappings with a graph
- Mixing coded systems
- Methods of comparison of families of real functions in porosity terms
