A NOTE ON ϱ-upper CONTINUOUS FUNCTIONS. 1. Preliminaries

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1 Tatra Mt Math Publ 44 (009), 5 58 DOI: 0478/v t m Mathematical Publicatios A NOTE ON ϱ-upper CONTINUOUS FUNCTIONS Stais law Kowalczyk Katarzya Nowakowska ABSTRACT I the preset paper, we itroduce the otio of classes of ϱ-upper cotiuous fuctios We show that ϱ-upper cotiuous fuctios are Lebesgue measurable ad, for ϱ<, may ot belog to Baire class We also prove that a fuctio with Deoy property ca be o-measurable Prelimiaries First, we shall collect some of the otios ad defiitios which appear frequetly i the sequel We apply stadard symbols ad otatio By R we deote the set of real umbers, by N we deote the set of positive itegers B deotes the set of all Baire class fuctios The symbol stads for the Lebesgue measure o the real lie ad also for the absolute value of a real umber Throughout the paper we cosider oly real-valued fuctios defied o a ope iterval Let E be a measurable subset of R ad let x R Theumbers ad d + (E,x) = lim if d + (E,x) = lim sup E [x, x + t] t E [x, x + t] t are called the right lower desity of E at x ad right upper desity of E at x The left lower ad upper desities of E at x are defied aalogously If d + (E,x) =d + (E,x) ad d (E,x) =d (E,x), 000 M a t h e m a t i c s Subect Classificatio: 6A5, 54C0 K e y w o r d s: desity of a set at a poit, cotiuous fuctios, measurable fuctios, path cotiuity, Deoy property, Baire class 5

2 STANIS LAW KOWALCZYK KATARZYNA NOWAKOWSKA the we call these umbers the right desity ad left desity of E at x, respectively The umbers ad d(e,x) = lim sup k 0 + t+k 0 d(e,x) = lim if k 0 + t+k 0 E [x t, x + k] k + t E [x t, x + k] k + t are called the upper ad lower desity of E at x, respectively If d(e,x)=d(e,x), we call this umber the desity of E at x ad deote it by d(e,x) Whe d(e,x) =,wesaythatx is a desity poit of E Defiitio Let E be a measurable subset of R Letx R ad 0 <ϱ< We say that the poit x is a poit of ϱ-type upper desity of E if d(e,x) >ϱ Defiitio A real-valued fuctio f defied o a ope iterval I is called ϱ-upper cotiuous at x provided that there is a measurable set E I such that x is a poit of ϱ-type upper desity of E, x E ad f E is cotiuous at x Iff is ϱ-upper cotiuous at every poit of I, we say that f is ϱ-upper cotiuous We will deote the class of all ϱ-upper cotiuous fuctios defied o a ope iterval I by UC ϱ The otio of ϱ-upper cotiuity is a example of so called path cotiuity, which was widely described i [] Mai results Theorem Let 0 <ϱ< Iff UC ϱ,thef is measurable Proof Let f : I R Assume that f UC ϱ ad suppose that f is ot measurable The there exists a umber a R for which the set { x I : f(x) <a } is o-measurable Deote A = { x I : f(x) <a }, B = { x I : f(x) a } It is obvious that B = I \ A is also o-measurable Cosider a measurable sets A A, B B such that A \ A ad B \ B do ot cotai a set of positive measure Therefore, A \ A ad B \ B are o-measurable sets If F =(A \ A ) (B \ B )=I \ (A B ), the F is a measurable set of positive measure Let L(F ) be a set of all desity poits of the set F By the well-kow Lebesgue Desity Theorem [], F \ L(F ) = 0 Therefore, there exists x 0 (A \ A ) L(F ) 54

3 ANOTEONϱ-UPPER CONTINUOUS FUNCTIONS Sice f is ϱ-upper cotiuous at x 0, it follows that there exists a measurable set E I such that x 0 E, d(e,x 0 ) >ϱad f E is cotiuous at x 0 As x 0 A, wehavef(x 0 ) <a Therefore, it is possible to fid δ>0 such that E (x 0 δ, x 0 + δ) A Let E = E (x 0 δ, x 0 + δ) Hece x 0 E, f E is cotiuous at x 0, E A ad d(e,x 0 )=d(e,x 0 ) >ϱ>0 () We have E =(E A ) ( E (A \ A ) ) Sice E ad E A are measurable, E (A \ A ) is measurable, too Hece, E (A \ A ) =0Moreover, d(e A,x 0 )= d ( ) I \ (E A ),x 0 d(f, x0 )= =0, because (E A ) F = It follows that d(e,x 0 ) d(e A,x 0 )+d ( ) E (A \ A ),x 0 =0+0=0, cotradictig () Thus, the assumptio that f may be o-measurable is false Defiitio We say that a real-valued fuctio f defied o a ope iterval I has Deoy property at x 0 I if for each ε>0adδ>0theset { x (x0 δ, x 0 +δ): f(x) f(x 0 ) <ε } cotais a measurable subset of positive measure We say that f has Deoy property if it has Deoy property at each poit x I Remark Let 0 <ϱ< If f UC ϱ the f has Deoy property Theorem There exists a o-measurable fuctio with Deoy property P r o o f First, we will costruct iductively a sequece {A : } of measurable sets such that () A i A = for each i, () A > 0 for each, () (a,b) R N A A A (a, b) Let { (a,b ) } be a sequece of all ope itervals with ratioal edpoits N Cosider (a,b ) There exists a closed iterval [α,β ] (a,b )adthereexist pairwise disoit closed subitervals I, I, I of [α,β ] Let C I be a perfect owhere dese set with positive measure such that C I I for =,, Let A = C I, A = C I, A = C I 55

4 STANIS LAW KOWALCZYK KATARZYNA NOWAKOWSKA Sice the set C I is owhere dese, there exists a closed iterval [α,β ] such that α <β ad [α,β ] (a,b ) \ (C I ) Moreover, oe ca fid pairwise disoit closed subitervals I, I, I of [α,β ] Let C I be a perfect owhere dese set with positive measure such that C I I for =,, Deote A 4 = C I, A 5 = C I, A 6 = C I Assume that sets the sets A,A,,A ( ) are already chose ad the coditios () ad () are fulfilled Also, assume that A i A i A i (a i,b i ) for every i Sice A A ( ) is a owhere dese set, there exists [α,β ] such that α <β ad [α,β ] (a,b ) \ ( A A ( ) ) Fix ay three closed pairwise disoit odegeerate itervals I, I, I [α,β ] LetC I be a Cator set with positive measure such that C I I for =,, Let A = C I, A = C I, A = C I So, by recursio, we ca costruct a sequece {A : } of measurable sets such that A i A = for each i, A > 0 for each ad A A A (a,b ) for every N Choose ay iterval (a, b) R Thereexists 0 such that (a 0,b 0 ) (a, b) Hece, A 0 A 0 A 0 (a 0,b 0 ) (a, b) Let B = A be ay o-measurable set Defie E = B = A ad let f : R R be a characteristic fuctio of the set E Let x R We cosider two cases If x E, thef(x) = 0 The for each ε>0wehave { } t: f(t) f(x) <ε A Sice (a, b) A > 0 = for each (a, b) R, wehavethatforeachε > 0adδ > 0theset { t (x δ, x + δ): f(t) f(x) <ε } cotais a measurable set of positive measure 56 =

5 ANOTEONϱ-UPPER CONTINUOUS FUNCTIONS If x E, thef(x) = The for each ε>0wehave { } t: f(t) f(x) <ε A Sice (a, b) A > 0 = for each (a, b) R, wehavethatforeachε > 0adδ > 0theset { t (x δ, x + δ): f(t) f(x) <ε } cotais a measurable set of positive measure Hece, f has Deoy property at x Sicex was arbitrary, we coclude that f has Deoy property Certaily, the set E is o-measurable ad f is o-measurable Theorem UC ϱ B for every ϱ< P r o o f We will costruct a fuctio f such that f UC ϱ for every ϱ< ad f is ot i Baire class Let C [0, ] be a perfect owhere dese set with positive measure such that for arbitrary iterval (a, b), if C (a, b), the C (a, b) > 0 Let A = = { x C : d(c, x) > From Lebesgue Desity Theorem, C \ A =0Iparticular,A is dese i C O the other had, eds of ambiguous itervals do ot belog to A Hece, the set C \ A is dese i C, tooletf : R R be a characteristic fuctio of the set A, f = χ A SiceA ad C \ A are dese i C, we deduce that f is discotiuous at each poit x C Hece, f is ot Baire fuctio We will show that f UC ϱ for every ϱ< If x A, thef A is costat, so it is cotiuous at x ad d(a, x) =d(c, x) d(c, x) >ϱ because C \ A =0 If x R\C, thef (R\C) is cotiuous at x ad R\C is a ope subset of R Hece, d(r \ C, x) =>ϱ If x C \ A, thef (R \ A) is cotiuous at x Sicex A, wehavethat d(c, x) It follows that } d(r \ A, x) = d(a, x) = d(c, x) = >ϱ Hece, f UC ϱ for every ϱ< 57

6 STANIS LAW KOWALCZYK KATARZYNA NOWAKOWSKA REFERENCES [] BRUCKNER, A M: Differetiatio of Real Fuctios, Lecture Notes i Math, Vol 659, Spriger-Verlag, New York, 978 [] BRUCKNER, A M O MALLEY, R J THOMSON, B S: Path derivatives: a uified view of certai geeralized derivatives, Tras Amer Math Soc 8 (984), 97 5 Received December 9, 008 Stais law Kowalczyk Istitute of Mathematics Academia Pomeraiesis ul Arciszewskiego b PL S lupsk POLAND stkowalcz@oeteu Katarzya Nowakowska Istitute of Mathematics Academia Pomeraiesis ul Arciszewskiego b PL S lupsk POLAND owakowska k@gopl 58