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ON THE SUMS OF UNILATERALLY CONTINUOUS JUMP FUNCTIONS

ON THE SUMS OF UNILATERALLY CONTINUOUS JUMP FUNCTIONS DEMONSTRATIO MATHEMATICAVol. XXXVNo 42002Marcin Grande*ON THE SUMSOF UNILATERALLY CONTINUOUS J U M P FUNCTIONSAbstract. In this article we prove that each jump function / : TZ —» TZ is the sumof two unilaterally continuous jump functions. Moreover we observe that there are jumpfunctions g : TZ —» TZ which are not the products of any finite family of unilaterallycontinuous jump functions.Let TZ be the set of all reals. A function / : TZ —> TZ is said to be a jumpfunction if for each point x ETZ there are the both finite unilateral limitsf(x+)= lim f ( t ) and / ( x - ) = lim /(t).t—>x+t-* X —It is known ([1]) that each jump function / may be discontinuous on countable set only, i.e. the set D ( f ) of all discontinuity points of / is countable.Observe thatR E M A R K 1. Let / : TZ —• TZ be a function. If for each point x G TZ there isat least one finite unilateral limit / ( x + ) or f(x—) then the set D ( f ) of alldiscontinuity points of / is countable.P r o o f . If x is http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ON THE SUMS OF UNILATERALLY CONTINUOUS JUMP FUNCTIONS

Demonstratio Mathematica , Volume 35 (4): 6 – Oct 1, 2002

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Publisher
de Gruyter
Copyright
© by Marcin Grande
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2002-0405
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXXVNo 42002Marcin Grande*ON THE SUMSOF UNILATERALLY CONTINUOUS J U M P FUNCTIONSAbstract. In this article we prove that each jump function / : TZ —» TZ is the sumof two unilaterally continuous jump functions. Moreover we observe that there are jumpfunctions g : TZ —» TZ which are not the products of any finite family of unilaterallycontinuous jump functions.Let TZ be the set of all reals. A function / : TZ —> TZ is said to be a jumpfunction if for each point x ETZ there are the both finite unilateral limitsf(x+)= lim f ( t ) and / ( x - ) = lim /(t).t—>x+t-* X —It is known ([1]) that each jump function / may be discontinuous on countable set only, i.e. the set D ( f ) of all discontinuity points of / is countable.Observe thatR E M A R K 1. Let / : TZ —• TZ be a function. If for each point x G TZ there isat least one finite unilateral limit / ( x + ) or f(x—) then the set D ( f ) of alldiscontinuity points of / is countable.P r o o f . If x is

Journal

Demonstratio Mathematicade Gruyter

Published: Oct 1, 2002

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