Around the Świątkowski-type conditions

• Autorzy: Filipczak Małgorzata , Terepeta Małgorzata

Identyfikatory

DOI

10.1515/jaa-2022-1043

• Języki publikacji: EN

Abstrakty

EN

In this paper, we will focus on different types of Świątkowski conditions: Świątkowski, strong Świątkowski and weak Świątkowski conditions. We present the main properties of the families of functions fulfilling such conditions.

Słowa kluczowe

EN

Świątkowski function; strong Świątkowski function; weakly Świątkowski function; cliquish function; quasi-continuity; algebrability; lineability; linear sensitivity

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2023
• Tom: Vol. 29, nr 2
• Strony: 205--217
• Opis fizyczny: Bibliogr. 44 poz.1 rys., portr.

Twórcy

autor

Filipczak Małgorzata

• Faculty of Mathematics and Computer Science, University of Lodz, ul. Stefana Banacha 22, 90-238 Łódź, Poland

• https://orcid.org/0000-0002-1137-9648

autor

Terepeta Małgorzata

• Centre of Mathematics and Physics and Institute of Mathematics, Lodz University of Technology, Żeromskiego 116, 90-924 Łódź, Poland

• https://orcid.org/0000-0001-5561-3398

Bibliografia

• [1] R. Aron, D. García and M. Maestre, Linearity in non-linear problems, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95 (2001), no. 1, 7-12.
• [2] R. Aron, V. I. Gurariy and J. B. Seoane, Lineability and spaceability of sets of functions on ℝ, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795-803.
• [3] R. M. Aron, L. Bernal González, D. M. Pellegrino and J. B. Seoane Sepúlveda, Lineability: The Search for Linearity in Mathematics, Monogr. Res. Notes Math., CRC Press, Boca Raton, 2016.
• [4] R. M. Aron and J. B. Seoane-Sepúlveda, Algebrability of the set of everywhere surjective functions on ℂ, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 25-31.
• [5] T. Banakh, M. G. Filipczak and J. Wódka, Returning functions with closed graph are continuous, Math. Slovaca 70 (2020), no. 2, 297-304.
• [6] A. Bartoszewicz, M. Bienias and S. Gła¸ b, Lineability, algebrability and strong algebrability of some sets in ℝℝ or ℂℂ, in: Traditional and Present-Day Topics in Real Analysis, University of Łódź, Łódź (2013), 213-232.
• [7] A. Bartoszewicz, M. Filipczak and M. Terepeta, Lineability of linearly sensitive functions, Results Math. 75 (2020), no. 2, Paper No. 64.
• [8] A. Bartoszewicz, M. Filipczak and M. Terepeta, On algebraic properties of the family of weakly Świątkowski functions, Results Math. 78 (2023), no. 3, Paper No. 87.
• [9] A. Bartoszewicz and S. Gła¸ b, Strong algebrability of sets of sequences and functions, Proc. Amer. Math. Soc. 141 (2013), no. 3, 827-835.
• [10] F. Bayart and L. Quarta, Algebras in sets of queer functions, Israel J. Math. 158 (2007), 285-296.
• [11] L. Bernal-González, D. Pellegrino and J. B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N. S.) 51 (2014), no. 1, 71-130.
• [12] A. M. Bruckner, J. G. Ceder and M. Weiss, Uniform limits of Darboux functions, Colloq. Math. 15 (1966), 65-77.
• [13] A. M. Bruckner and J. L. Leonard, Derivatives, Amer. Math. Monthly 73 (1966), no. 4, 24-56.
• [14] E. P. Dolženko, Boundary properties of arbitrary functions, Math. USSR Izv. 31 (1967), 3-14.
• [15] 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.
• [16] Z. Grande, On a subclass of the family of Darboux functions, Colloq. Math. 117 (2009), no. 1, 95-104.
• [17] G. Ivanova and A. Karasińska, About porosity of some Świątkowski functions in the space of quasi-continuous functions, Quaest. Math. 44 (2021), no. 1, 37-43.
• [18] G. Ivanova and E. Wagner-Bojakowska, On some modification of Świątkowski property, Tatra Mount. 58 (2014), 101-109.
• [19] G. Ivanova and E. Wagner-Bojakowska, On some modification of Darboux property, Math. Slovaca 66 (2016), no. 1, 79-88.
• [20] G. Ivanova and E. Wagner-Bojakowska, Porous subsets in the space of functions having the Baire property, Math. Slovaca 67 (2017), no. 6, 1333-1344.
• [21] I. Jóźwik and M. Terepeta, Profesor Tadeusz Świątkowski - dobry duch Politechniki Łódzkiej, Uniw. Jana Długosza w Częstochowie Mathematics 19 (2014), 277-286.
• [22] S. Kempisty, Sur les fonctions quasicontinues, Fund. Math. 19 (1932), 184-197.
• [23] E. Kocela and T. Świątkowski, On some characterization of uniform convergence, Zeszyty Nauk. Politech. Łódz. Mat. (1976), no. 7, 11-15.
• [24] J. Kucner, Funkcje posiadające silną własność Świątkowskiego, PhD thesis, Łódź, 2002.
• [25] A. Maliszewski, On the limits of strong Świątkowski function, Zeszyty Nauk. Politech. Łódz. Mat. (1995), no. 27, 87-93.
• [26] A. Maliszewski, Sums and products of quasi-continuous functions, Real Anal. Exchange 21 (1995/96), no. 1, 320-329.
• [27] A. Maliszewski, Darboux Property and Quasi-continuity: A Uniform Approach, Wydaw. Uczelniane WSP, Warsaw, 1996.
• [28] T. Mańk and T. Świątkowski, On some class of functions with Darboux’s characteristic, Zeszyty Nauk. Politech. Łódz. Mat. (1978), no. 11, 5-10.
• [29] M. Marciniak and P. Szczuka, On internally strong Świa¸ tkowski functions, Real Anal. Exchange 38 (2012/13), no. 2, 259-272.
• [30] R. Menkyna, On the sums of lower semicontinuous strong Świątkowski functions, Real Anal. Exchange 39 (2013/14), no. 1, 15-32.
• [31] T. Natkaniec and J. Wódka, On the pointwise limits of sequences of Świątkowski functions, Czechoslovak Math. J. 68(143) (2018), no. 4, 875-888.
• [32] C. J. Neugebauer, Darboux property for functions of several variables, Trans. Amer. Math. Soc. 107 (1963), 30-37.
• [33] H. Pawlak and R. Pawlak, On some conditions equivalent to the condition of Świątkowski for Darboux functions of one and two variables, Zeszyty Nauk. Politech. Łódz. Mat. 16 (1983), 33-40.
• [34] H. Pawlak and W. Wilczyński, On the condition of Darboux and Świątkowski for functions of two variables, Zeszyty Nauk. Politech. Łódz. Mat. 15 (1982), 31-35.
• [35] R. Pawlak, Przekształcenia Darboux, Uniwersytet Łódzki, Łódź, 1985.
• [36] R. J. Pawlak, On Świątkowski functions, Acta Univ. Lodz. Folia Math. 3 (1989), 77-85.
• [37] T. Radaković, Über Darbouxsche und stetige Funktionen, Monatsh. Math. Phys. 38 (1931), no. 1, 117-122.
• [38] T. Świątkowski, On the conditions of monotonicity of functions, Fund. Math. 59 (1966), 189-201.
• [39] P. Szczuka, Maximal classes for the family of strong Świątkowski functions, Real Anal. Exchange 28 (2002/03), no. 2, 429-437.
• [40] H. P. Thielman, Types of functions, Amer. Math. Monthly 60 (1953), 156-161.
• [41] J. Wódka, Subsets of some families of real functions and their algebrability, Linear Algebra Appl. 459 (2014), 454-464.
• [42] J. Wódka, On the uniform limits of sequences of Świątkowski functions, Lith. Math. J. 57 (2017), no. 2, 259-265.
• [43] Z. Zahorski, Sur la première dérivée, Trans. Amer. Math. Soc. 69 (1950), 1-54.
• [44] L. Zajíček, Porosity and σ-porosity, Real Anal. Exchange 13 (1987/88), no. 2, 314-350.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).