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A NOTE ON THE GRAPH QUASICONTINUITY

A NOTE ON THE GRAPH QUASICONTINUITY DEMONSTRATE MATHEMATICAVol. XXXIXNo 32006Zbigniew GrandeA NOTE ON THE G R A P H QUASICONTINUITYAbstract. Let (X,px) and ( Y , p y ) be metric spaces. A function / : X —• Y is saidgraph quasicontinuous if there is a quasicontinuous function g : X —> Y with the graphGr(g) contained in the closure cl(Gr(f)) of Gr(f). If the space ( Y , p y ) is compact and ifthere is a dense subset A C X such that the restricted function / / A is continuous then/ is graph quasicontinuous. Moreover each locally bounded function / : R —• R is graphquasicontinuous.Let ( X , px) and (Y, py) be metric spaces. A function / : X —• Y is saidto be quasicontinuous at a point x € X ([4]) if for each real rj > 0 and foreach open neighbourhood U 3 x there is a nonempty open set V C U suchthat /(V) C K(f(x),rf), where K(f(x),rj) denotes the open ball with thecenter f(x) and the radius 77. The function / is said quasicontinuous if it isquasicontinuous at each point x E X.A function / : X —> Y is said to be graph quasicontinuous ([2]) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

A NOTE ON THE GRAPH QUASICONTINUITY

Grande, Zbigniew
Demonstratio Mathematica , Volume 39 (3): 4 – Jul 1, 2006
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References (2)

Publisher
de Gruyter
Copyright
© by Zbigniew Grande
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2006-0305
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATE MATHEMATICAVol. XXXIXNo 32006Zbigniew GrandeA NOTE ON THE G R A P H QUASICONTINUITYAbstract. Let (X,px) and ( Y , p y ) be metric spaces. A function / : X —• Y is saidgraph quasicontinuous if there is a quasicontinuous function g : X —> Y with the graphGr(g) contained in the closure cl(Gr(f)) of Gr(f). If the space ( Y , p y ) is compact and ifthere is a dense subset A C X such that the restricted function / / A is continuous then/ is graph quasicontinuous. Moreover each locally bounded function / : R —• R is graphquasicontinuous.Let ( X , px) and (Y, py) be metric spaces. A function / : X —• Y is saidto be quasicontinuous at a point x € X ([4]) if for each real rj > 0 and foreach open neighbourhood U 3 x there is a nonempty open set V C U suchthat /(V) C K(f(x),rf), where K(f(x),rj) denotes the open ball with thecenter f(x) and the radius 77. The function / is said quasicontinuous if it isquasicontinuous at each point x E X.A function / : X —> Y is said to be graph quasicontinuous ([2])

Journal

Demonstratio Mathematica – de Gruyter

Published: Jul 1, 2006

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