SUMS AND PRODUCTS OF EXTRA STRONG ŚWIA TKOWSKI FUNCTIONS. 1. Preliminaries

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1 Ø Ñ Å Ø Ñ Ø Ð ÈÙ Ð Ø ÓÒ DOI: /v Tatra Mt. Math. Publ. 49 (2011), SUMS AND PRODUCTS OF EXTRA STRONG ŚWIA TKOWSKI FUNCTIONS Paulina Szczuka ABSTRACT. In this paper we present a characterization of sums of extra strong Świ atkowski functions, and we examine some functions which can be written as the product of extra strong Świ atkowski functions. 1. Preliminaries The letters R and N denote the real line and the set of positive integers, respectively. The word function means a mapping from R into R. The symbols I(a,b) and I[a,b] denote the open and the closed interval with endpoints a and b, respectively. For each A R we use the symbols IntA, cla, bda, and A to denote the interior, the closure, the boundary, and the outer Lebesgue measure of A, respectively. The Euclidean metric in R will be denoted by ρ. We say that a set A R is simply open [1], if it can be written as the union of an open set and a nowhere dense set. Let f be a function. The symbol C(f) stands for the set of all points of continuity of f. If A R and x is a limit point of A, then let lim(f,a,x) = lim t x,t A f(t). Similarly, we define lim(f,a,x + ), lim(f,a,x + ), etc. Moreover, we write lim(f,x) insteadoflim(f,r,x),etc.setc = lim(f,x )andd = lim(f,x ).Wesaythatx R is a Darboux point of f from the left if c f(x) d and f [ (x δ,x) ] (c,d) for each δ > 0. Similarly, we define the notion of a Darboux point from the right. WesaythatxisaDarbouxpointof f (x D(f))ifxisaDarbouxpointoff both c 2011 Mathematical Institute, Slovak Academy of Sciences Mathematics Subject Classification: Primary 26A21, 54C30; Secondary 26A15, 54C08. Keywords: Darboux function, quasi-continuous function, strong Świ atkowskifunction,extra strong Świ atkowski function, sum of functions, product of functions. Supported by Kazimierz Wielki University. 71

2 PAULINA SZCZUKA from the left and from the right. Recall that f is a Darboux function 1 (f D) if and only if each x R is a Darboux point of f. (See, e.g., [3, Theorem 5.1].) We say that f is quasi-continuous in the sense of Kempisty [5] (f Q), if for all x R and open sets U x and V f(x), the set Int ( U f 1 (V) ) is non- -empty. We say that f is cliquish [15] (f C q ), if the set of points of continuity of f is dense in R. We say that f is a strong Świ atkowski function [6] (f Śs), if whenever α,β R and y I ( f(α),f(β) ), there is an x 0 I(α,β) C(f) such that f(x 0 ) = y. We say that f is an extra strong Świ atkowski function (f Śes), if whenever α,β R, α β, and y I [ f(α),f(β) ], there is anx 0 I[α,β] C(f) such that f(x 0 ) = y. Remark 1.1º We can easily see that the following inclusions are satisfied Ś es Śs DQ D and DQ Q C q. We say that a function f changes its sign in interval I, if there are points x 1,x 2 I such that sgnf(x 1 ) sgnf(x 2 ). The symbol M denotes the class of all functions f such that f has a zero in each interval in which it takes on both positive and negative values. The symbols [f = a] and [f a] stand for the sets x R : f(x) = a } and x R : f(x) a }, respectively. 2. Introduction In 1996 Maliszewski proved the following theorem [9]. Ì ÓÖ Ñ 2.1º For each function f the following conditions are equivalent: i) f is a finite product of Darboux quasi-continuous functions, ii) there are Darboux quasi-continuous functions g and h such that f = gh, iii) f M, f is cliquish, and the set [f = 0] is simply open. He showed also that products of two and three strong Świ atkowski functions are different, and asked for characterization of products of such functions. In 2006 I found a partial solution to this problem proving the following theorem [14]. Ì ÓÖ Ñ 2.2º For each function f the following conditions are equivalent: i) f is a finite product of strong Świ atkowski functions, ii) there are strong Świ atkowski functions g 1,...,g 4 such that f = g 1...g 4, iii) the function f is cliquish, the set [f = 0] is simply open, and there exists a G δ -set A [f = 0] such that I A for every interval I in which f takes on both positive and negative values. 1 We say that f is a Darboux function if it maps connected sets onto connected sets. 72

3 SUMS AND PRODUCTS OF EXTRA STRONG ŚWIA TKOWSKI FUNCTIONS In this paper we examine the smaller class of functions, namely the family Śes of extra strong Świ atkowski functions. We give equivalent conditions of belonging to the class Śes (Theorem 3.1). It turns out that using one of them and the proof of Maliszewski theorem [7, Theorem 4], we obtain directly the characterization of sums of extra strong Świ atkowski functions. Moreover, we examine some functions which can be written as the product of two, three, four, and as the finite product of extra strong Świ atkowski functions. 3. Sums First we will give equivalent conditions of belonging to the class Śes. Ì ÓÖ Ñ 3.1º For each function f the following conditions are equivalent: i) f Śes, ii) f D and f[i] = f [ I C(f) ] for each nondegenerate interval I, iii) f D and f(x) f [ I[x,t] C(f) ] for each x R and each t R\x}. Proof. i) ii) By Remark 1.1 we clearly have f D. Assume that y f[i]. Then f(x) = y for some x I. There is a compact interval J I such that x J. Since y I [ f(minj),f(maxj) ], there is an x 0 J C(f) I C(f) with f(x 0 ) = y. We proved that f[i] = f [ I C(f) ]. ii) iii) Let x,t R and t x. Then obviously f(x) f [ I[x,t] ] = f [ I[x,t] C(f) ]. iii) i) Let α,β R, α β and y I [ f(α),f(β) ]. Since f D, we have y = f(x y ) for some x y I[α,β]. Let t I[α,β]\x y }. By assumption y f [ I[x y,t] C(f) ], whence f(x 0 ) = y for some x 0 I[x y,t] C(f) I[α,β] C(f). An immediate consequence of Theorem 3.1 is a local characterization of extra strong Świ atkowski functions. ÓÖÓÐÐ ÖÝ 3.2º For each function f the following conditions are equivalent: i) f Śes, ( ii) ( x R) x D(f) ( t R) ( t x f(x) f[i[x,t] C(f)] )). Ì ÓÖ Ñ 3.3º Let f 1,f 2,...,f k be cliquishfunctions. There is a functiong such that f i +g is an extra strong Świ atkowski function for each i 1,2,...,k}. 73

4 PAULINA SZCZUKA The proof of Theorem 3.3 is almost the same as the proof of [7, Theorem 4], which was presented by Maliszewski in Using Theorem 3.1, immediately from condition (8) of [7], we obtain that functions f 1 +g,f 2 +g,...,f k +g are extra strong Świ atkowski. ÓÖÓÐÐ ÖÝ 3.4º If a function f is cliquish, then there are extra strong Świ atkowski functions h 1,h 2 such that f = h 1 +h 2. Proof. Use Theorem 3.3 for the family f,0}. Using the above corollary, Remark 1.1, and [8, Corollary 3.2] we can easily prove the following theorem. Ì ÓÖ Ñ 3.5º For every function f the following conditions are equivalent: i) f is a finite sum of extra strong Świ atkowski functions, ii) there are extra strong Świ atkowski functions h 1and h 2 such that f=h 1 +h 2, iii) f is cliquish. 4. Products In 1960 S. Marcus remarked that not every function is the product of Darboux functions [10]. In 1985 Z. Grande constructed a nonnegative Baire one function which cannot be the product of a finite number of quasi-continuous functions [4]. Eleven years later A. Maliszewski proved that there is a bounded Darboux quasi-continuous function which cannot be written as the finite product of strong Świ atkowski functions [9]. Now we will show that there is a strong Świ atkowski function which cannot be written as the finite product of extra strong Świ atkowski functions. Ì ÓÖ Ñ 4.1º If function f is a finite product of extra strong Świ atkowski functions, then f is cliquish, the set [f = 0] is simply open, and there is a G δ -set A [f = 0] such that I A for every interval I in which the function f changes its sign. Proof. Assume that there is a k N such that f = g 1...g k with g 1,...,g k Ś es.since Śes Śs,by Theorem2.2, thefunctionf iscliquish andthe set[f = 0] is simply open (cf. also [11, Theorem] or [2]). For i 1,...,k}, define g i = min maxg i, 1},1 }. Then the function g i is bounded. By [12, Corollary 3.6], g i Śes. 74

5 SUMS AND PRODUCTS OF EXTRA STRONG ŚWIA TKOWSKI FUNCTIONS Put f = g 1... g k. Observe that sgn f = sgn f. Define a G δ -set A = Int f 1( ( n 1,n 1)) = [ f = 0] C( f) [f = 0]. n=1 Let I R be an interval in which f changes its sign. Then at least one of the functions g 1,..., g k, say g 1, changes its sign in I, too. Since g 1 Śes, there is an x 0 I C( g 1 ) such that g 1 (x 0 ) = 0. The functions g 1,..., g k are bounded, so x 0 C( f), and finally x 0 I [ f = 0] C( f) = I A. ÈÖÓÔÓ Ø ÓÒ 4.2º There is a nonnegative bounded strong Świ atkowski function which cannot be written as the finite product of extra strong Świ atkowski functions. Proof. Let F be a Cantor ternary set, and let I be a family of all components of the set R\F. Define ρ(x,f) / I if x cli, I I, f(x) = 1/2 otherwise. It can be readily verified that f is nonnegative bounded and Darboux, f cli is continuous, and f[i] = (0,1/2] for all I I. Since C(f) = R \ F and f[f] = 0,1/2}, we have f Śs. Now suppose that there is a G δ -set A [f = 0] such that I A for every interval I in which the function f changes its sign. By definition of f we have A I IbdI and cla = F. Hence A is at most countable, dense in F, G δ -set, a contradiction. By Theorem 4.1 the function f cannot be written as the finite product of extra strong Świ atkowski functions. All functions which can be written as the finite product of Darboux, (quasi- -continuous, Darboux quasi-continuous) functions are the product of two Darboux (quasi-continuous, Darboux quasi-continuous) functions, respectively (see, [9, TheoremsIII.1.4, III.2.1, and III.3.1]). In 1996 A. Maliszewski remarked that the sign function can be written as the product of three strong Świ atkowski functions but it cannot be factored into the product of two such functions [9, Propositions III.4.2 and III.4.3], and asked for characterization of products of strong Świ atkowski functions. 75

6 PAULINA SZCZUKA In 2003 I found a function which is the product of four strong Świ atkowski functions, and which cannot be written as the product of three strong Świ atkowski functions [13, Example]. Three years later I proved Theorem 2.2 [14, Theorem 4.2]. However, the problem of characterization of products of two and three strong Świ atkowski functions is still open. The next two assertions show that products of two and three extra strong Świ atkowski functions are different. Ä ÑÑ 4.3º If a function f can be written as a product of two extra strong Świ atkowski functions, then for each interval I, if f changes its sign in I, for some x I [f = 0]. lim ( f,c(f),x ) = lim ( f,c(f),x +) = 0 Proof. Suppose that f = g 1 g 2, where g 1,g 2 Śes. If f changes its sign in an interval I, then either g 1 or g 2, say g 1, has the same property. Since the function g 1 Śes, there is an x I C(g 1 ) such that g 1 (x) = 0. Clearly lim ( g 1,C(g 1 ),x ) = 0. Since g 2 Śes, then lim ( g 2,C(g 2 ),x ) < and lim ( g 2,C(g 2 ),x +) <. As C(g 1 ) and C(g 2 ) are residual, so the set C(g 1 ) C(g 2 ) is dense. Hence lim ( g 1 g 2,C(g 1 ) C(g 2 ),x ) = lim ( g 1 g 2,C(g 1 ) C(g 2 ),x +) = 0. Obviously, C(g 1 ) C(g 2 ) C(f) and we obtain lim ( f,c(f),x ) = lim ( f,c(f),x +) = 0. ÈÖÓÔÓ Ø ÓÒ 4.4º There is a nonnegative bounded cliquish function which can be written as the product of three extra strong Świ atkowski functions, and which cannot be written as the product of two extra strong Świ atkowski functions. Proof. Let f(x) = sgnx. Then f is nonnegative, bounded, and cliquish. Define g 1 (x) = x, x +1+sinx 1 if x 0, h(x) = 0 if x = 0, h(x)/ x if x 0, g 2 (x) = 2 if x = 0, and g 3 (x) = 1/h(x) if x 0, 2 if x = 0. 76

7 SUMS AND PRODUCTS OF EXTRA STRONG ŚWIA TKOWSKI FUNCTIONS Obviously, g 1 g 2 g 3 = sgn = f, the function g 1 is continuous, whence extra strong Świ atkowski, and C(g 2) = C(g 3 ) = R\0}. Since lim(g 2,0) = lim(g 3,0) =, lim(g 2,0) = 1, lim(g 3,0) = 1/2 and g 2 (0) = g 3 (0) = 2, then g 2,g 3 D, and by Theorem 3.1, g 2,g 3 Śes. So, the function f = sgn can be written as the product of three extra strong Świ atkowski functions. Now observe that for each x R lim ( f,c(f),x ) = lim ( f,c(f),x +) = 1 0. Hence, by Lemma 4.3, the function f = sgn cannot be written as the product of two extra strong Świ atkowski functions. Now we will show that products of three and four extra strong Świ atkowski functions are different. ÈÖÓÔÓ Ø ÓÒ 4.5º There is a bounded cliquish function f which can be written as the product of four extra strong Świ atkowski functions, and which cannot be written as the product of three extra strong Świ atkowski functions. Proof. Define 0 if x = (nπ) 1, n N, f(x) = 1 if x = 0, 1 otherwise. Obviously, the function f is bounded. Since the set C(f) = R\ ( [f = 0] 0} ) is dense in R, the function f is cliquish. For i 1,2} define sinx 1 if x 0, g i (x) = ( 1) i if x = 0. Clearly, g 1,g 2 Śes and sgn f = sgn (g 1 g 2 ). Now define We can easily see that and f(x) = sin 2 x 1 if f(x) 0, 1 otherwise. f = f/(g 1 g 2 ) on the set [f 0] C( f) = R\ ( [f = 0] 0} ). 77

8 PAULINA SZCZUKA Obviously, C( f) is dense in R, whence f is cliquish. Moreover, f > 0 on R and ln f is cliquish, too. By Corollary 3.4, there are extra strong Świ atkowski functions h 1,h 2 such that ln f = h 1 +h 2. So, f = exp (h 1 +h 2 ) = (exp h 1 )(exp h 2 ). Define g 3 = exp h 1 and g 4 = exp h 2. By [12, Lemma 2.3], g 3,g 4 Śes. Consequently, f = g 1...g 4, where g 1,...,g 4 Śes. Now suppose that there are functions g 1,g 2,g 3 Śes such that f = g 1 g 2 g 3. Then sgn f = sgn (g 1 g 2 g 3 ). Note that f takes on both positive and negative values in each right-hand neighborhood of 0. So at least one of the functions g 1,g 2,g 3, say g 1, has the same property in each right-hand neighborhood of 0. Assume that g 1 (0) < 0. (The case g 1 (0) > 0 is similar.) There is an x 0 (0,π 1 ) such that g 1 (x 0 ) > 0. Let z = sup x < x 0 : g 1 (x) < 0 }. We have 0 z x 0. If g 1 (z) < 0, then z < x 0 and g 1 (x) 0 for each x (z,x 0 ]. In this case g 1 / D Śes, a contradiction. So, g 1 (z) 0, whence z > 0. By definition, we have z x 0 π 1, and z ( (nπ + π) 1,(nπ) 1] for some n N. If g 1 (z) > 0, then there is a z 1 ( (nπ + π) 1,z ) such that g 1 (z 1 ) < 0. But f 0 on ( (nπ + π) 1,(nπ) 1). So, g 1 0on(z 1,z)andfinallyg 1 / D Śes,acontradiction.Thereforeweconclude that g 1 (z) = 0, whence z = (nπ) 1 and z < x 0 < π 1. Since g 1 (z 1 ) < 0 for some z 1 ( (nπ + π) 1,z ), g 1 > 0 on ( z,(nπ π) 1), and g 1 Śes, we obtain z C(g 1 ). Clearly, sgn g 1 = sgn (g 2 g 3 ) on ( (nπ +π) 1,z ) ( z,(nπ π) 1). Since the function g 1 takes on both positive and negative values in every neighborhood of z, one of the functions g 2 and g 3, say g 2, has the same property in every neighborhood of z. But g 2 Śes, whence g 2 (z) = 0 and z C(g 2 ). Finally observe that lim ( g 3,z +) = lim ( f/(g 1 g 2 ),z +) =. Consequently, g 3 / D Śes, contrary to our assumption. We have shown that thefunctionf cannotbewrittenastheproductofthreeextrastrongświ atkowski functions. Finally we would like to present the problem. ÈÖÓ Ð Ñ 4.6º Characterize products of extra strong Świ atkowski functions. 78

9 SUMS AND PRODUCTS OF EXTRA STRONG ŚWIA TKOWSKI FUNCTIONS REFERENCES [1] BISWAS, N.: On some mappings in topological spaces, Bull. Calcutta Math. Soc. 61 (1969), [2] BORSÍK,J.:Products of simply continuous and quasicontinuous functions,math.slovaca 45 (1995), [3] BRUCKNER, A. M. CEDER, J. G.: Darboux continuity, Jahresber. Deutsch. Math.- -Verein. 67 (1965), [4] GRANDE, Z.: Sur les fonctions cliquish, Časopis Pěst. Mat. 110 (1985), [5] KEMPISTY, S.: Sur les fonctions quasicontinues, Fund. Math. 19 (1932), [6] MALISZEWSKI, A.: On the limits of strong Świ atkowski functions, Zeszyty Nauk. Politech. Lódz. Mat. 27 (1995), [7] MALISZEWSKI, A.: On theorems of Pu & Pu and Grande, Math. Bohem. 121 (1996), [8] MALISZEWSKI, A.: On the sums and the products of Darboux cliquish functions, Acta Math. Hungar. 73 (1996), [9] MALISZEWSKI, A.: Darboux Property and Quasi-Continuity. A Uniform Approach, WSP, S lupsk, [10] MARCUS, S.: Sur la représentation d une fonction arbitraire par des fonctions jouissant de la propriété de Darboux, Trans. Amer. Math. Soc. 95 (1960), [11] NATKANIEC, T.: Products of quasi-continuous functions, Math. Slovaca 40 (1990), [12] SZCZUKA, P.: Maximal classes for the family of strong Świ atkowski functions, Real Anal. Exchange 28 (2002/03), [13] SZCZUKA, P.: Products of strong Świ atkowski functions, in: Proc. of Internat. Conf. on Real Functions Theory, Rowy, [14] SZCZUKA, P.: Products of strong Świ atkowski functions, J. Appl. Anal. 12 (2006), [15] THIELMAN, H. P.: Types of functions, Amer. Math. Monthly 60 (1953), Received November 15, 2010 Kazimierz Wielki University pl. Weyssenhoffa 11 PL Bydgoszcz POLAND paulinaszczuka@wp.pl 79