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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
Demonstratio Mathematica – de Gruyter
Published: Oct 1, 2002
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