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MAXIMAL CLASSES FOR THE FAMILY OF QUASI-CONTINUOUS FUNCTIONS WITH CLOSED GRAPH

MAXIMAL CLASSES FOR THE FAMILY OF QUASI-CONTINUOUS FUNCTIONS WITH CLOSED GRAPH DEMONSTRATIO MATHEMATICANo 12009Vol. XLIIWaldemar SiegMAXIMAL CLASSES FOR THE FAMILY OFQUASI-CONTINUOUS FUNCTIONS WITH CLOSED G R A P HAbstract. In this paper we consider classes of functions / : R —» K. The maximaladditive class for the family QjU of quasi-continuous functions with closed graph is equalto the class of all continuous functions. We also show that the maximal multiplicativeclass for QJA is equal to a class of continuous functions, which fulfil an extra condition.1. IntroductionThrough out this paper R denotes the set of all real numbers, and weconsider R and R x R endowed with their natural topologies. The symbol R Kstands for the set of all functions / : R —> R, and the symbols C, Const,Q,V, B\ and U denote the subsets of R® consisting of all continuous, constant,quasi-continuous, Darboux, Baire-one and functions with closed graph, respectively. Moreover, we setC* ={/ GC:/ = 0 orf(x) ±0,forWe will also use the following abbreviations.For T and Q nonempty subsets of R r , the symboland the setsTQall xG R}.denotes the setJ-C\Q,>R•MaCn = {<? e R e : (Vf e F) g + f e J7},Mm(Jr)= { g € R=are calledmaximalthe familyadditive,of functions:: (V/ e F ) m a x http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

MAXIMAL CLASSES FOR THE FAMILY OF QUASI-CONTINUOUS FUNCTIONS WITH CLOSED GRAPH

Sieg, Waldemar
Demonstratio Mathematica , Volume 42 (1): 6 – Jan 1, 2009
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References (10)

Publisher
de Gruyter
Copyright
© by Waldemar Sieg
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2009-0104
Publisher site
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Abstract

DEMONSTRATIO MATHEMATICANo 12009Vol. XLIIWaldemar SiegMAXIMAL CLASSES FOR THE FAMILY OFQUASI-CONTINUOUS FUNCTIONS WITH CLOSED G R A P HAbstract. In this paper we consider classes of functions / : R —» K. The maximaladditive class for the family QjU of quasi-continuous functions with closed graph is equalto the class of all continuous functions. We also show that the maximal multiplicativeclass for QJA is equal to a class of continuous functions, which fulfil an extra condition.1. IntroductionThrough out this paper R denotes the set of all real numbers, and weconsider R and R x R endowed with their natural topologies. The symbol R Kstands for the set of all functions / : R —> R, and the symbols C, Const,Q,V, B\ and U denote the subsets of R® consisting of all continuous, constant,quasi-continuous, Darboux, Baire-one and functions with closed graph, respectively. Moreover, we setC* ={/ GC:/ = 0 orf(x) ±0,forWe will also use the following abbreviations.For T and Q nonempty subsets of R r , the symboland the setsTQall xG R}.denotes the setJ-C\Q,>R•MaCn = {<? e R e : (Vf e F) g + f e J7},Mm(Jr)= { g € R=are calledmaximalthe familyadditive,of functions:: (V/ e F ) m a x

Journal

Demonstratio Mathematica – de Gruyter

Published: Jan 1, 2009

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