Czechoslovak Mathematical Journal, Vol. 68, No. 4, pp. 875-888, 2018


On the pointwise limits of sequences of Świątkowski functions

Tomasz Natkaniec, Julia Wódka

Received January 17, 2017.   Published online May 4, 2018.

Abstract:  The characterization of the pointwise limits of the sequences of Świątkowski functions is given. Modifications of Świątkowski property with respect to different topologies finer than the Euclidean topology are discussed.
Keywords:  Świątkowski function; cliquish function; pointwise limit; $^\ast$topology of Hashimoto; $\mathcal{I}$-density topology; density topology
Classification MSC:  26A21, 26A15, 54C08, 54C30


References:
[1] M. Filipczak, G. Ivanova, J Wódka: Comparison of some families of real functions in porosity terms. Math. Slovaca 67 (2017), 1155-1164. DOI 10.1515/ms-2017-0039 | MR 3707118 | Zbl 06786764
[2] Z. Grande: Sur la quasi-continuité et la quasi-continuité approximative. Fundam. Math. 129 (1988), 167-172. (In French.) DOI 10.4064/fm-129-3-167-172 | MR 0962538 | Zbl 0657.26003
[3] Z. Grande: On a subclass of the family of Darboux functions. Colloq. Math. 117 (2009), 95-104. DOI 10.4064/cm117-1-6 | MR 2539550 | Zbl 1177.26005
[4] H. Hashimoto: On the *topology and its application. Fundam. Math. 91 (1976), 5-10. DOI 10.4064/fm-91-1-5-10 | MR 0413058 | Zbl 0357.54002
[5] G. Ivanova, E. Z. Wagner-Bojakowska: On some subclasses of the family of Darboux Baire 1 functions. Opusc. Math. 34 (2014), 777-788. DOI 10.7494/OpMath.2014.34.4.777 | MR 3283017 | Zbl 1339.26013
[6] G. Ivanova, E. Z. Wagner-Bojakowska: On some modification of Darboux property. Math. Slovaca 66 (2016), 79-88. DOI 10.1515/ms-2015-0117 | MR 3510852 | Zbl 06589831
[7] M. Kowalewski, P. Szczuka: Separating sets by Świątkowski functions. Quaest. Math. 39 (2016), 471-477. DOI 10.2989/16073606.2015.1096857 | MR 3521167
[8] C. Kuratowski: Topologie. Vol. I. Panstwowe Wydawnictwo Naukowe, Warszawa (1958). (In French.) MR 0090795 | Zbl 0078.14603
[9] R. Lester: Pointwise Discontinuous Function. Dissertation, University of Missouri, Missouri (1912).
[10] A. Maliszewski: On the limits of strong Świątkowski function. Zesz. Nauk. Politech. Lódź. 719, Mat. 27 (1995), 87-93. MR 1357159 | Zbl 0885.26002
[11] A. Maliszewski: Darboux Property and Quasi-Continuity. A Uniform Approach. Dissertation, WSP, Słupsk (1996).
[12] A. Maliszewski, J. Wódka: Products of Świątkowski and quasi-continuous functions. J. Appl. Anal. 20 (2014), 129-132. DOI 10.1515/jaa-2014-0014 | MR 3284719 | Zbl 1305.26012
[13] A. Maliszewski, J. Wódka: Products of Świątkowski functions. Math. Slovaca 66 (2016), 601-604. DOI 10.1515/ms-2015-0163 | MR 3543724 | Zbl 06639573
[14] T. Mańk, T. Świątkowski: On some class of functions with Darboux's characteristic. Zesz. Nauk. Politech. Lodz., Mat. 11 (1978), 5-10. MR 3472109 | Zbl 0416.26005
[15] M. Marciniak, P. Szczuka: $A$-Darboux functions. Lith. Math. J. 56 (2016), 107-113. DOI 10.1007/s10986-016-9307-2 | MR 3472109 | Zbl 1342.26011
[16] J. C. Oxtoby: Measure and Category. A Survey of the Analogies between Topological and Measure Spaces. Graduate Texts in Mathematics 2, Springer, New York (1980). DOI 10.1007/978-1-4684-9339-9 | MR 0584443 | Zbl 0435.28011
[17] H. Pawlak, R. Pawlak: On some conditions equivalent to the condition of Świątkowski for Darboux functions of one and two variables. Zesz. Nauk., Politech. Lódź. 413, Mat. 16 (1983), 33-40. MR 0744182 | Zbl 0597.26003
[18] R. J. Pawlak: Przekształcenia Darboux. Dissertation, Lódź University (1985). (In Polish.)
[19] W. Poreda, E. Wagner-Bojakowska, W. Wilczyński: A category analogue of the density topology. Fundam. Math. 125 (1985), 167-173. DOI 10.4064/fm-125-2-167-173 | MR 0813753 | Zbl 0613.26002
[20] H. P. Thielman: Types of functions. Am. Math. Mon. 60 (1953), 156-161. DOI 10.2307/2307568 | MR 0052495 | Zbl 0051.13801
[21] J. Wódka: Subsets of some families of real functions and their algebrability. Linear Algebra Appl. 459 (2014), 454-464. DOI 10.1016/j.laa.2014.07.015 | MR 3247237 | Zbl 1309.15005
[22] J. Wódka: On the uniform limits of sequences of Świątkowski functions. Lith. Math. J. 57 (2017), 259-265. DOI 10.1007/s10986-017-9359-y | MR 3654989 | Zbl 06767581

Affiliations:   Tomasz Natkaniec, Institute of Mathematics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland, e-mail: Tomasz.Natkaniec@mat.ug.edu.pl; Julia Wódka, Lódź University of Technology, Institute of Mathematics, ul. Wólczańska 21, 90-924 Lódź, Poland, e-mail: JuliaWodka@gmail.com


 
PDF available at: