On universal elements for some families of functions

• Autorzy: Grande Z. , Strońska E.
• Języki publikacji: EN

Abstrakty

EN

A point x C X is called universal element for a family phi of functions from X to y if the set {f(x)\f 6 phi} is dense in Y. In this article we show that every residual G- set in a completely regular space X (every residual set in R ) is the set of all universal elements for some family of continuous functions from X to R (for some family of quasicontinuous functions from Rk to R). Moreover we investigate the sets of all universal elements for some families of monotone functions and for some families of functions having the property of Denjoy-Clarkson.

Słowa kluczowe

EN

universal element; quasicontinuity; density topology; monotone function; residual set

PL

elementy uniwersalne; funkcje monotoniczne; topologia gęstości

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2004
• Tom: Vol. 37, nr 3
• Strony: 679--688
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Grande Z.

• The University of Computer Science and Economics TWP, ul. Wyzwolenia 30, 10-106 Olsztyn, Poland

autor

Strońska E.

• Institute of Mathematics, Bydgoszcz Academy, Plac Weyssenhoffa 11, 85-072 Bydgoszcz, Poland

Bibliografia

• [1] A. M. Bruckner, Differentiation of Real Functions, Lectures Notes in Math. 659, Springer-Verlag, Berlin 1978.
• [2] J. A. Clarkson, A property of derivatives, Bull. Amer. Math. Soc. 53 (1947), 124-125.
• [3] A. Denjoy, Sur les fonctions dérivées summables, Bull. Soc. Math. France 43 (1915), 161-248.
• [4] R. Engelking, General Topology, (Polish) PWN, Warszawa (1989), vol. 1.
• [5] Z. Grande, Sur les fonctions approximativement quasi-continues, Rev. Roum. Math. Pures et Appl. 34 (1989), 17-22.
• [6] K. G. Grosse-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt, Math. Sem. Giessen 176 (1987), 1 -84.
• [7] K. G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999), 345-381.
• [8] G. Herzog, R. Lemmert, Universal elements for families of discontinuous mappings, Demonstratio Math. 35 (2002), 199-204.
• [9] I. Joó, On the divergence of eigenfunction expansion, Ann. Univ. Sci. Budapest Eotvos Sect. Math. 32 (1989), 3-36.
• [10] F. B. Jones, Connected and disconected plane sets and the functional equations f(x+y) = f(x) + f(y), Bull. Amer. Math. Sci. 48, (1942), 115-120.
• [11] S. Kempisty, Sur les fonctions quasicontinues, Fund. Math. 19 (1932), 184-197.
• [12] T. Neubrunn, Quasi-continuity, Real Anal. Exch. 14 (1988-89), 259-306.
• [13] F. D. Tall, The density topology, Pacific J. Math. 62 (1976), 275-284.
• [14] W. Wilczyński, Density Topologies, Edited by Pap E. Handbook of Measure Theory, Elsevier Science B.V., Amsterdam (2002), vol. 1, chap. XV.