Natkaniec, Tomasz

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Products of quasi-continuous functions

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Mathematica SlovacaTomasz NatkaniecProducts of quasi-continuous functionsMathematica Slovaca, Vol. 40 (1990), No. 4, 401--405Persistent URL: http://dml.cz/dmlcz/129385Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMath . Slovaca 40, 1990, No. 4, 401—405 PRODUCTS OF QUASI-CONTINUOUS FUNCTIONS TOMASZ NATKANIEC ABSTRACT. Functions which can be factored into a product of quasi-continuous functions are characterized. Let us establish some of the terminology to be used. R denotes the real line. Forf: R -> IR and aelRwe define the set [f< a] as {xGlR:f(x) < a}. Analo-gously we define the sets [f> a] and [f= a]. A real function f defined on IR is said to be quasi-continuous (cliquish) at a point x0e IR iff for every s > 0 and for any neighbourhood [/of the point x0 there exists an open set V such that 0 9-= V a U and |f(x) — f(x0)l < £ f° r e a c h xeV (respectively |f(x,) — f(x2)l < £ for x]9 x2eV) [I]. A function f is quasi-continuous (cliquish) on an interval J c R iff f is quasi-continuous (cliquish) at every point of J. Of course, every point at which fis continuous is a quasi-continuity point off and every quasi-continuity point off is a cliquish point of this function. - It is easy to see that a sum or a product of two cliquish functions are cliquish. Thus a sum and a product of quasi-continuous functions are cliquish (but not necessarily quasi-continuous). Z. G r a n d e proved in [2] (Th. 5) that every cliquish function fis a sum of four quasi-continuous functions. Observe that if fis Lebesgue measurable (or with the Baire property, or of the Baire class a)9 then the factors from the proof of Grande can be taken from the adequate class. In fact Grande proved his theorem for cliquish functions defined on IR but from that result it follows easily that every cliquish function defined on an open interval / is also sum of four quasi-continuous functions (defined on I). Now we may notice the following fact. Fact. If g and h are quasi-continuous on an open interval /and a9beU9 then there exists a function s continuous on / such that g + s and h — s are quasi-continuous functions whose cluster sets at the left and the right end points of / contain a and b9 respectively. AMS Subject Classif icat ion (1985): Primary 26A16 Key words: Quasi-continuity, Cliquish, Lebesque measurable, Baire property, Baire class 401 Of course, if g and h are Lebesgue measurable (or with the Baire property, or of the Baire class a), then g + s and h - s are of the adequate class too. Consequently, Grande's theorem holds for cliquish functions defined on any interval of an arbitrary type. In the same paper ([2]) Grande remarked that the function/: U -• U defined as J 1/q if x is rational, x = P/q, q > 0 and (p, q) = 1, { 0 if x is irrational cannot be the product of a finite number of quasi-continuous functions. The purpose of this article is to characterize those functions which can be factored into a product of quasi-continuous functions. The result is as follows. Theorem. A function h: R -> IR is a product of quasi-continuous functions iff: (i) h is cliquish, and (ii) each of the sets [h = 0], [h > 0] and [h < 0] is the union of an open set and a nowhere-dense set. Moreover, ifh is Lebesgue measurable (respectively with the Baire property or of Baire class a), then the factors can be taken to be Lebesgue measurable (respectively with the Baire property or of the Baire class a). We begin with the following lemma. Lemma. If I a R is an interval and f: /-> IR is a cliquish function which is always positive or always negative, then f is the product of four quasi-continuous functions/, . . . , / : /->IR. Moreover, if f is Lebesgue measurable (or with the Baire property, or of the Baire class a), then f, . . . , / can be taken from the adequate class. Proof. Let g(x) = | /(x) | for xel. Then ln(g) is a cliquish function and there exist quasi-continuous functions gx, ..., g4: I-+M such that \n(g) = = g\ + gi + gi + gA [2]. Now the functions/ = sgn (/) • exp (gx),f = exp (g) for i = 2, 3, 4, are quasi-continuous a n d / = / °... °/4. The second part of this lemma follows from the remark that i f / i s of the adequate class, then In (g) is of this same class and gu ...,g4 belong to this class too. P roof of Theorem. First, let us assume that h is a product of n quasi-continuous functions h = / ° . . . o / w . h, as a product of cliquish functions, is cliquish. Notice that for a quasi-continuous function/each of the sets [ /= 0], [ /< 0] and [ /> 0] is the union of an open set G(f 0), G(f - ) and G(f, +) (respective-ly) and a nowhere-dense set A(f 0), A(f —) and A(f +). Here we can assume that the set A(f 0) is disjoint with G(f 0) and analogously for A(f - ) and n n A(f,+). Thus [h = 0] = ( J G(fh 0) u \J A{f„ 0) and this set is the union of 402 an open set (J {G(f, 0) / = 1, 2, ..., n} and nowhere-dense set (J [A(f, 0): / = 1, 2, ..., n). Since [A < 0] = [f • ... -f_ , < 0] n [f > 0] u [f o... ofn_ 1 > 0] n n [f < 0], we obtain (by induction) that this set is the union of an open set and a nowhere-dense set. Similarly we can prove that the set [A > 0] is the union of an open set and a nowhere-dense set. Let us then assume that A satisfies the conditions (i) and (ii). The set A = (G(A, 0)\G(A, 0))u(G(A, +)\G(A, + ) ) u u (G(A, -) \G(A, - ) ) = IR \ (G(A, 0) u G(A, + ) u G(A, - ) ) = A(h9 0) u u A(h, +) u A(h, —) is closed and nowhere-dense and there exists a sequence (Kn)n of non-empty open intervals such that: (1) KnnA = 0, i.e. Kn a G(A, 0), or Kn cz G(A, +) , or Kn a G(A, - ) , (2) if Kn n Km ^ 0, then n = ra, n,me N, (3) for every xeA and for each neighbourhood U of x the set {n: Kn a U} is infinite, (4) if x$A, then there exists a neighbourhood U of x such that the set {n: Un Kn^ 0} has at most one element. Fix n e N. We can choose a sequence (Knm)m ^,. of open subintervals of Kn such that: (5) Knm cz K„ for m ^ /i, (6) if K„9m n KnJ # 0, then m = t. Notice that the set ^ u G(A, 0)u (J (J K„f,„ is closed. Let (/„)„ be a n — \ m = 1 sequence of all components of the complement of this set. Then In a G(A, —) or / „ c G(A, + ) for every component In. Let tn], ..., tn4: /„-*IR be quasi-continuous functions such that A | /„ = twj °... ° tM. Similarly, if the interval Kw m is not contained in the set G(A, 0), then Kn m cz G(A, —) or Knm a G(A, +) and there exist quasi-continuous functions Km,u •••> KmX- ^ m - ^ R s u c h t h a t h\&n,m is the product of A:nfMf„ ..., fc„iWf4. Let (w„)„ be a sequence of all rationals different than zero. Now we can define quasi-continuous functionsf, ...,f8: IR -• IR in the following way. for xeA, for x€K„t2m9 n ^ 2m, /.(*) = < 0 for x6G(/7, 0) \UK„ , 2m? »,i- + i.iW for xeKn<2m+l\G(h, 0), n^2m+l, for XGL, «eI^J 403 j for x e A u \J KnЛm + ,, n,m (Уwjkм(x) forxeЌя.2m\G(h,0), n>lm, fi(x) = { 0 f0Г xeG(h, 0)\ U K„.2m+ „ w.m 1 for xe\jln n 1 for xeA u U £..2*,u G(A> 0), / i ( 1 ľ s1 *«.2-. + ,.,-.(*) f°r ^eK, JГ,\G(Л, 0), „ > 2itt + 1, 1 for x e A u U K».2m +i u G(A, 0), jj(*) = < ^ . ^ - 4 ^ ) for xeK,2 w\G(/i,0), « ^ 2 m , 1 for x e U I > -У= 6,7,8 Clearly /? = f o... o/8. Let us now show thatf is quasi-continuous. The proof thatf, i = 2, ..., 8, are quasi-continuous is similar. It is enough to verify that f, is quasi-continuous at every point x0eA. Fix e > 0 and a neighbourhood J7 of x0. Let wm be a rational number from the interval (h(x0) — £, h(x0) + e). Then there exists n^ 2m such that A 2̂m c t/ andf(x) -= wm for jceK„%7m- Thusf, is quasi-continuous at the point JC0. If h is a function of the Baire class a (Lebesgue measurable, with the Baire property), then the functionsf, ...,f8 belong to the adequate class. We shall verify this fact forf in the case when h is a Baire 1 function. Then every function tni is of the Baire class 1 on In (ne N, i = 1, 2, 3, 4) and every function kn m , is of the Baire class 1 on Knm (neN,m^n,i= 1, 2, 3, 4). For an open set G c U each of the sets h~](G)nA} []{Kn^: wmeG\ ( J k~l + u(G)n (K2m+l\G(h,0)) and (J t~l(G) n I„ is a Fa set. Moreover, the set ( J Knlm n G(h, 0) is closed in n n,m G(h, 0), thus the set G(h, 0)\ \J K„ 2m is open. It follows that/ ,- 1 (G) is a Fa set, n,m so f is of the Baire class 1. This finishes the proof of the Theorem. 404 REFERENCES [1] BLEDSOҚ W. W.: Neighbourly functions, Proc. Amer. Math. Soc. 3, 1952, 114—115. [2] GRANDE, Z.: Sur les functions cliquish. Časop. Pěst. Mat. 110, 1985, 225—236. Received June 22, 1989 Instytut Matematyki WSP ul. Chodkiewicza 30 85 064 Bydgoszcz Poland 405 

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Products of quasi-continuous functions

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