Mathematica Bohemica

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1 Mathematica Bohemica Jonald P. Fenecios; Emmanuel A. Cabral; Abraham P. Racca Baire one functions and their sets of discontinuity Mathematica Bohemica, Vol. 141 (2016), No. 1, Persistent URL: Terms of use: Institute of Mathematics AS CR, 2016 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 141(2016) MATHEMATICA BOHEMICA No. 1, BAIRE ONE FUNCTIONS AND THEIR SETS OF DISCONTINUITY Jonald P. Fenecios, Davao City, Emmanuel A. Cabral, Quezon City, Abraham P. Racca, Silang Received January 25, 2014 Communicated by Jiří Spurný Abstract. A characterization of functions in the first Baire class in terms of their sets of discontinuityisgiven.moreprecisely,afunction f: R RisofthefirstBaireclassifand onlyifforeach ε >0thereisasequenceofclosedsets {C n} suchthat D f = C n and ω f (C n) < εforeach nwhere ω f (C n)=sup{ f(x) f(y) : x,y C n} and D f denotesthesetofpointsofdiscontinuityof f. Theproofofthemaintheorem isbasedonarecent ε-δcharacterizationofbaireclassonefunctionsaswellasonawellknown theorem due to Lebesgue. Some direct applications of the theorem are discussed in the paper. Keywords: Baire class one function; set of points of discontinuity; oscillation of a function MSC2010:26A21 1. Introduction Afunction f: R RisBaireclassoneoroffirstBaireclassorsimplyBaireone ifitisapointwiselimitofasequenceofcontinuousfunctionon R.HenriLebesgue showedin1904thatafunctionisofthefirstbaireclassifandonlyifforeach k N,thedomaincanberepresentedasacountableunionofclosedsetssothat theoscillationof foneachsetisstrictlylessthan 1/k,see[2],page116. Foreasy reference, we shall call this theorem Lebesgue s theorem. In the process it was proved thatthesetofpointsofdiscontinuityof fisasetofthefirstcategory. Fromthis, aquestionemerges: Doesafunctionwhosesetofpointsofdiscontinuityisofthe firstcategoryhavetobeabaireclassonefunction? Itturnsouttheanswerisno. DOI: /MB

3 Thereisafunction f: R Rwhosesetofpointsofdiscontinuityisofthefirst categoryandatthesametimeoflebesguemeasure0butwhichisnotbaireclass one. Hence, from this perspective it is hard to obtain a characterization of Baire classonefunctionsbothintermsofthecategoryandmeasureofitssetofpointsof discontinuity. However, a natural problem arises: can one still obtain a characterization of Baire classonefunctionsintermsoftheirsetofpointsofdiscontinuity? Weanswerthis question in the affirmative. 2. A new characterization Throughoutthepaper,welet C f and D f denotethesetofpointsofcontinuity and the set of points of discontinuity of f, respectively. Before presenting the main result, we need the following useful propositions. Proposition2.1([8]). If R = E n witheach E n an F σ in Rthenthereare disjoint F σ sets F n, n = 1,2,...in Rsuchthat F n E n and R = Proposition2.2([8]). Let R = F n. F n where F n saredisjoint F σ sets. Then thereisapositivefunction δ( )on Rsuchthat x F n, y F m and n mimply x y min{δ(x),δ(y)}. Weshallnowproveourmainresult. Theorem 2.1. Let f: R R. The following statements are equivalent: (1) fisbaireclassone. (2) Foreach ε > 0thereisasequenceofclosedsets {C n }suchthat D f = and ω f (C n ) < εforeach nwhere ω f (C n ) = sup{ f(x) f(y) : x,y C n }. C n Proof. (1) (2). Let ε > 0begiven. ByLebesgue stheorem,thereexists asequenceofclosedsets {E n } suchthat R = E n and ω f (E n ) < εforeach n. 110

4 Since D f isknowntobean F σ thenthereisasequenceofclosedsets {F n } such that D f = F n.itfollowsthatwecanexpress D f as D f = (E i F j ). i,j N Clearly, ω f (E i F j ) < εforanypair (i,j). (2) (1).WewillusethecharacterizationofBaireclassonefunctionduetoLee, TangandZhao[6]toestablish(1). Let ε > 0. Byassumption,thereisasequence ofclosedsets {C n }suchthat D f = C n and ω f (C n ) < εforeach n. Foreach x C f thereisacorrespondingpositivenumber δ x > 0suchthat y (x δ x,x+δ x ) f(x) f(y) < ε 2. Let C = {G: G = (x δ x,x + δ x )and x C f }. Then C f acountablesubcollection {G n }of Csuchthat C f ( R = Byreindexing,wecanwrite R = i=1 ( G i ) j=1 C j ). G C G n.itfollowsthat G. Wecanfind E n where E n = G i forsome ior E n = C j for some j.byproposition2.1wecanfindadisjointsequenceof F σ sets {F n }suchthat R = F n and F n E n foreach n.byproposition2.2thereisapositivefunction δ: R R + suchthat x F m, y F n with m nimplies x y min{δ(x),δ(y)}. Let x,y Rand x y < min{δ(x),δ(y)}.bythepropertyofthepositivefunction δ( )thereisaunique nsuchthat x,y F n.since F n E n implies f(x) f(y) < ε, alltheseshowthat fisbaireclassone. Remark. Thetheoremissayingthattodecidewhetherafunctionbelongs tothefirstbaireclass,onenolongerneedstoexaminethewholedomainofthe function, as Lebesgue s theorem is suggesting, but one examines instead the set of pointsofdiscontinuityofthefunction.inthissense,theorem2.1maybeviewedas an improvement of Lebesgue s theorem. ItmaybeobservedthatTheorem2.1canbeexpressedinaslightlydifferent mannerthatmayproveusefulinsomecases.wewillstateitasacorollary. 111

5 Corollary 2.1. Let f: R R. The following statements are equivalent: (1) fisbaireclassone. (2) Foreach ε > 0thereisasequenceofclosedsets {C n }suchthat D f and ω f (C n ) < εforeach nwhere ω f (C n ) = sup{ f(x) f(y) : x,y C n }. C n 3. Some applications Inthissection,weshalltrytogivesomeapplicationsofTheorem2.1.Ashortand quick proof that a function with countable set of discontinuity is Baire class one is perhapsthroughatheoremduetorenébaire:afunction f: R RisBaireclass oneifandonlyifforeveryclosedset K,therestriction f K hasapointofcontinuity in K. However, the statement above also admits a straighforward justification using Theorem2.1.Forotherproofsonemaysee[3],[4],[5],[7]. Theorem3.1.Let f: R R.If D f iscountablethen fisbaireclassone. Proof. Notethatacountablesetisacountableunionofsingletons.Clearlythe oscillation of the function on each singleton is zero. Theorem3.2.Let f: R R.If f(d f )iscountableandforeach y f(d f )the set {x D f : f(x) = y}is F σ then fisbaireclassone. Proof. Let f(d f ) = {r 1,r 2,...,r n,...}andlet F i = {x D f : f(x) = r i }for each i. Byassumption,each F i is F σ. Notethat D f F i and ω f (F i ) = 0for each i.bycorollary2.1, fisbaireclassone. Theorem3.3. Let f,g: R Rsuchthat D f D g.if gisbaireclassoneand ω f (A) ω g (A)forevery A D f then fisbaireclassone. Proof. Let ε > 0begiven.Since gisbaireclassonethenthereexistsasequence ofclosedsets {D n }suchthat D g = D n and ω g (D n ) < εforall n.since D f D g and D f is F σ thenwecanfindacountablecollection {E n }ofclosedsetssuchthat D f = E n and ω g (E n ) < εforeach n. Fromthehypothesis,itimmediately followsthat ω f (E n ) ω g (E n ) < εforeach n.thus, fisbaireclassone. 112 i=1

6 Lastly,wewillshowthatiff Df iscontinuousond f then fisbaireclassone.this classoffunctionsiscalled B1,see[1]. Moreover,weusethenewcharacterization toshowthatifthereisasequenceofclosedsets {E n } suchthat R = E n and therestriction f En iscontinuouson E n foreach nthen fisbaireclassone. Such functions are known as piecewise continuous functions. We need first the following lemma. Lemma3.1. Let Kbeaclosedsubsetof R. If f K iscontinuouson Kthen foreach ε > 0thereexistsasequence {K n }ofclosedsetscovering K satisfying ω f (K n ) < εforeach n. Proof. Let ε > 0.Foreach x Kthereisanopeninterval I x containing xsuch that y I x K f(x) f(y) < ε 2. Notethatthecollection C = {I x K: x K}formsanopencoverof Kwhere K isviewedasasubspaceof R. Since K isalindelöfsubspaceof R,wecanfind a countable subcollection of C such that K = (I ξi K). i=1 Notethat ω f (I ξi K) < εandeach I ξi Kis F σ.thelemmafollows. Theorem3.4.Let f: R R.If f Df iscontinuouson D f then fisbaireclass one. Proof. Since D f is F σ thenthereexistsasequenceofclosedsets {K n }in R suchthat D f = K n.since f Df iscontinuous, f Kn iscontinuousforeach n.we applylemma3.1andtheorem2.1toconcludethat fisbaireclassone. Theorem3.5.Let f: R R.Ifthereisasequenceofclosedsets {E n } such that R = E n andtherestriction f En iscontinuouson E n foreach nthen fis Baire class one. Proof. Let ε > 0begiven. Since {E n }covers Ritcertainlycovers D f. Since f En iscontinuousforeach nbylemma3.1thereexistsasequenceofclosedsets {F n }suchthat D f F n and ω f (F n ) < εforeach n.thusbycorollary2.1, fis Baire class one. 113

7 Itisalsointerestingtonotethatifthereisasequence {D n }ofclosedsetssuch that D f = D n and f Dn iscontinuousforeach nthen fisbaireoneonthewhole of R. Clearly, this class of functions lies between the class of all piecewise continuous functionsandtheclassofbaireonefunctions. Itisnotclearwhetherthisclassof functionsisthesameastheclassofallbaireonefunctions. Acknowledgement. Wewouldliketothanktherefereeforreadingcarefully our manuscript. References [1] A. Bąkowska, R. J. Pawlak: On some characterizations of Baire class one functions and Baire class one like functions. Tatra Mt. Math. Publ. 46(2010), [2] D. M. Bressoud: A Radical Approach to Lebesgue s Theory of Integration. MAA Textbooks, Cambridge University Press, Cambridge, [3] A. M. Bruckner, J. B. Bruckner, B. S. Thomson: Real Analysis. Prentice-Hall International, Upper Saddle River, [4] R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics 4, American Mathematical Society, Providence, [5] K. Kuratowski: Topology. I. Academic Press, New York; Państwowe Wydawnictwo Naukowe, Warszawa; Mir, Moskva, 1966.(In Russian.) [6] P.-Y.Lee,W.-K.Tang,D.Zhao:AnequivalentdefinitionoffunctionsofthefirstBaire class. Proc. Am. Math. Soc. 129(2001), [7] I. P. Natanson: Theory of Functions of a Real Variable. II. Frederick Ungar Publishing, New York, 1961.(In German.) [8] D. Zhao: Functions whose composition with Baire class one functions are Baire class one. Soochow J. Math. 33(2007), Authors addresses: Jonald P. Fenecios, Department of Mathematics, Ateneo de Davao University, E. Jacinto Street, 8016 Davao, Philippines, jpfenecios@addu.edu.ph; Emmanuel A. Cabral, Department of Mathematics, Ateneo de Manila University, Loyola Heights Campus, Katipunan Avenue, 1108 Quezon, Philippines, ecabral@ateneo. edu; Abraham P. Racca, Department of Mathematics and Physics, Adventist University of the Philippines, Puting Kahoy, Silang, 4118 Cavite, Philippines, abraham.racca@ yahoo.com. 114