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1 t m Mathematical Publications DOI: /tmmp Tatra Mt. Math. Publ. 68 (2017), A NOTE TO THE SIERPI SKI FIRST CLASS OF FUNCTIONS Robert Menkyna ABSTRACT. The purpose of this paper is to establish some theorems concerning approximation and representation of a function of the Sierpi ski first class by Darboux function of the Sierpi ski first class. In recent years, a number of articles have appeared dealing with a problem of approximating or representing arbitrary real valued functions of real variable by functions with the Darboux (intermediate value) property. In some cases, only functions belonging to a restricted class of functions such as the Baire one class [2], [3], [13] or the class of semicontinuous functions [6], [10], [12] were considered. We deal with the classes of real functions defined on the interval I =[0, 1]. First, we introduce all relevant notations. Let C, B 1, D, Q, Ś s, lsc, and usc stand for the class of continuous, Baire one, Darboux, quasi- -continuous, strong wi tkowski, lower and upper semicontinuous functions, respectively. The intersection D lsc will be denoted by Dlsc, and applying the same principle, we will use notations DQlsc, Ś s lsc, too.let C f be asetof all points of continuity of the function f, and D f be a set of all points of discontinuity of the function f. A point x is said to be a bilateral c-point ofaseta if and only if (x; x+δ) A and (x δ; x) A have the cardinality continuum for every δ>0, i.e., card ( (x; x + δ) A ) =card ( (x δ; x) A ) = c. The set A is said to be bilaterally c-dense in the set B, B c A,ifand only if each point x B is a bilateral c-point of the set A. If a function f : I R maps connected sets onto connected sets, then it is said to be Darboux. c 2017Mathematical Institute, Slovak Academy of Sciences Mathematics Subject Classification: Primary 26A21, 54C30; Secondary 26A15, 54C08. Keywords: lower resp. upper semicontinuity, the Darboux property, the strong wiatkowski function. This work was partially supported by the Slovak Grant Agency under the project VEGA No. 1/0676/17. 59

2 ROBERT MENKYNA Let us recall the notion of quasi-continuity which was introduced by S.Kempisty in 1932[4]. A function f is said to be quasi-continuous atapoint x 0 if for each open set U x 0 and each open set V f(x 0 ) there exists a nonempty open set W U such thatf(w ) V. According to [8], a function f is strong wi tkowski if whenever α, β I, α<βand y ( f(α),f(β) ), there exists a point x 0 (α, β) C f such that f(x 0 )=y. Now, we define the Sierpi ski first class S 1 of functions [15]: { } S 1 = f n : f n (x) < for every x [0, 1], with each f n C. n=1 n=1 According to [16], the Sierpi ski first class coincides with the class of sums of lower semicontinuous and upper semicontinuous functions. For this reason, the Sierpi ski first class will be denoted either by S 1 or, in accordance with [11], by lsc lsc. Itis evident that S 1 B 1 but S 1 B 1 (see [11], [16]). To prove the main results of the paper, we will use of [10, Theorem 3] on approximation of semicontinuous function by Darboux semicontinuous function. Theorem 1. Let f be a function such that f lsc, and let E be an arbitrary F σ set which is bilaterally c-dense in itself. If the set E is bilaterally c-dense in the set of points of discontinuity of the function f, then there exists a function g Dlsc such that g(x) <f(x) for x E, g(x) =f(x) for x I \ E. Now, we prove the theorem on approximation by Darboux function in the Sierpi ski first class. Theorem 2. Let f S 1, f = l + u, where l lsc and u usc. IfaBorel set E is bilaterally c-dense in the set of points of discontinuity { of functions } l and u, then there exists a function g DS 1 such that x: f(x) g(x) E. Proof. We denote D = D l D u. The set D is of type F σ, of first category, that is, D = n=1 D n,whered 1 D 2 D 3 are closed nowhere dense sets and, of course, D f D.Additionally, according to [13, Lemma 7], it can be assumed that the set E is of type F σ, of the first category, bilaterally c-dense in itself and n=1 D n c E. Otherwise, the set E can be replaced with its subset having these properties. Thus, let E = F n, n=1 where each F n is a nowhere dense closed set. Without loss of generality,itcanbe 60

3 A NOTE TO THE SIERPI SKI FIRST CLASS OF FUNCTIONS assumed that F 1 F 2 As l lsc, there exists a sequence of continuous functions l 1 <l 2 <l 3 < which pointwise converges to the function l and, analogously, as u usc, there exists a sequence of continuous functions u 1 >u 2 >u 3 > which pointwise converges to the function u. Evidently, the sequence f n = l n + u n, n N, pointwise converges to the function f. Moreover, let {ε n } n N be a sequence of positive realnumbers such that ε n 0. Functions l n,u n,f n are uniformly continuous on [0, 1]. Thus, the sequence {ε n } n N determines a sequence of positive numbers δ n such that for every x 1,x 2 [0, 1] x 1 x 2 <δ n l n (x 1 ) l n (x 2 ) <ε n, u n (x 1 ) u n (x 2 ) <ε n, f n (x 1 ) f n (x 2 ) <ε n. (0.1) Applying of [9, Lemma 2], we can construct a sequence of perfect sets P n, n N, such that F 1 c P 1 E and (F n+1 P n ) c P n+1 E for every n N. It is evident that P 1 c P 2 c P 3 c, E = P i. Let for each n N the triple of functions (l n,u n,f n ) be associated with the perfect set P n. The set E is c-dense in D and therefore F n can be required to satisfy condition x D n there is a, b F n P n such that a<x<b b a<δ n. (0.2) From the existence of a system of perfect sets P n, n = 1, 2,..., it follows that there exists a system of closed sets P α,α 1 (proof in [13, Theorem 8]) such thatif α 1 <α 2 P α1 c P α2. Now, for each i α<i+1,we define a triple of functions (l α,u α,f α ), associated with the set P α. i=1 l α = l i + (α i)(l i +1 l i ), u α = u i + (α i)(u i+1 u i ), f α = f i + (α i)(f i+1 f i ), 61

4 ROBERT MENKYNA It is easy to see that if α 1 <α 2 l α1 < l α2, u α1 > u α2 and f α = l α + u α for each α 1. Let α (x) have the same sense as in the proof of [10, Theorem 3] if: x E α (x) =inf{α: x P α }. Now, we define the functions l,u,f in the way asin the proof in [10, Theorem 3]: { l l α(x) (x), if x E, (x) = l (x), if x/ E, { u u α(x) (x), if x E, (x) = u (x), if x/ E, { f f (x) = α(x) (x), if x E, f (x), if x/ E. The function l fulfills identical conditions as the function g in the proof of [10, Theorem 3], that is l Dlsc and l <l on E l = l on [0, 1] \ E. A theorem analogous with Theorem 1 can be formulated for the function u usc, too. Therefore, the function u Dusc and u >u on E, u = u on [0, 1] \ E. Moreover, the function f is constructed in the same way as the function g in [13, Theorem 8]. According to the assertion of this theorem, f DB 1 and { x: f (x) f(x) } E. Because f α = l α +u α for every α 1, from the construction of functions l,u,f it follows that f = l + u S 1 B 1. Using the last two results, we show that the function f { DB 1 S 1 = DS 1 and for x: f (x) f(x) } E, that is, the function f satisfies the assertion of theorem. The next theorem proves that every Sierpi ski one function can be represented as a sum of two Sierpi ski one functions with Darboux property. 62

5 A NOTE TO THE SIERPI SKI FIRST CLASS OF FUNCTIONS Theorem 3. Let f be a Sierpínski one function on an interval I, f = l + u, where l lsc and u usc. IfaBorel set E C l C u is bilaterally c-dense in the set of points of discontinuity of the functions l and u, then there exists a function g DS 1 such that for { x: f(x) g(x) } E and the function f g DS 1. Proof. Again, by [13, Lemma 7], there exists a set E E C f of type F σ of the first category, bilaterally c-dense in itself such that D l D u c E. Theorem 2 implies the existence of a function g DS 1, g = l + u, where l lsc and u usc such that the set for { x: f(x) g(x) } E. Because the function f g =(l + u) (l + u )=(l u )+(u l ) S 1, it suffices to prove that the function f g has Darboux property. Let us consider an arbitrary point x 0 I. If f(x 0 )=g(x 0 ), then since the set for { x : f(x) =g(x) } is residual in I, there exist sequences x n x 0, y n x 0, n N, f(x n )=g(x n ), f(y n )=g(y n ). Thus, lim (f g)(x n) = lim (f g)(y n)=(f g)(x 0 )=0. n n If f(x 0 ) g(x 0 ), then x 0 C f. The function g DS 1, hence there exist sequences x n x 0, y n x 0, n N, such that lim g (x n) = lim g (y n)=g (x 0 ). n n The function f is continuous at the point x 0. Then, From the foregoing, it follows that lim f (x n) = lim f (y n)=f (x 0 ). n n lim (f g)(x n) = lim (f g)(y n)=(f g)(x 0 ). n n Following Young in [1], f g DB 1. From the above, it follows: { } f g DB 1 S 1 = DS 1 and for x; g(x) f(x) E. Of course, if we choose a first Baire category set E of Lebesque measure zero from Theorem 3, then there exists a Darboux Sierpi ski one function g such that f = g +(f g), where f g DS 1 and f g =0except for on the first category measure zero set. On the other hand, if we do not require validity of f g =0except for on the measure zero set, then we can show that every Sierpi ski one function can be expressed as a sum of two Sierpi ski one functions with a stronger property than the Darboux property. Based on results of A. Maliszewski [7], 63

6 ROBERT MENKYNA R. Men ky n a [14]characterized the class of differences of lower semicontinuous strong wi tkowski functions by equations (lsc lsc) DQ = lscdq lscdq = Śslsc Śslsc. Additionally, in[12], it is proved that lsc = Śslsc + Śslsc or usc = Śsusc + Śsusc, respectively. If we use the validity of the above relations, we can prove the following statement: Theorem 4. Let f be a Sierpi ski one function on an interval I. Then, there exist Darboux quasi-continuous Sierpi ski one functions f 1,f 2 defined on the interval I such that f = f 1 + f 2. P r o o f. Because f is the Sierpi ski one function, there exists a lower semicontinuous function l and a upper semicontinuous function u such that f = l + u. In accord with [12, Theorem 8], there exist lower semicontinuous strong wi tkowski functions l 1,l 2 and upper semicontinuous strong wi tkowski functions u 1,u 2 such that l = l 1 + l 2 and u = u 1 + u 2. Therefore, f =(l 1 + l 2 )+(u 1 + u 2 )= ( l 1 ( u 1 ) ) + ( l 2 ( u 2 ) ). The function f 1 = l 1 ( u 1 ) Śslsc Śslsc = lscdq lscdq =(lsc lsc) DQ = DQS 1 and for the same reason, the function f 2 = l 2 ( u 2 ) DQS 1. We have shown that any Sierpi ski one function can be written as a sum of two Darboux quasi-continuous Sierpi ski one functions. In addition, our goal is to prove that every Sierpi ski one function can be expressed as a sum of two strong wi tkowski functions of Sierpi ski one class. By small modifications to the proof of Lemma A.3.2 and to the proof of the implication ii iii in the proof of [5, Theorem A.3.3], we can prove the folowing lemmas. Naturally, let symbols lim (f,x), lim (f,x), g,ω(f,x) have the same meaning as in [5]. Lemma 5. Assume that K is nowhere dense and closed, the functions f 1,f 2,......,f m DQB 1 are locally bounded onr \ K, and τ>0. There is a nowhere dense closed set F K such that F \ K m j=1 C f j,andcontinuous function g, g =0on K, g τ such that (f j + g)(i F \ K) [ lim (f j,x), lim (f j,x) ] for every x K, j =1, 2,...,m and every open interval I x. 64

7 A NOTE TO THE SIERPI SKI FIRST CLASS OF FUNCTIONS Proof. Write R\K as the union of a sequence of compact intervals {I n,n N} with boundary points from m j=1 C f j such that int(i n ) int(i m )=, forevery n m.for each n,putc j n =inf f j (I n ) and L j n =supf j (I n ).Let{J p,p N} be an enumeration of all open intervals with rational endpoints which intersect K. (The lemma is trivial in case K =.) Fix p N. First, observe that for each k 0 N, j =1, 2,...,m (c j n τ 2 p,lj n + τ ) [ cl 2 lim (fj,x), lim (f p j,x) ]. (0.3) x K J p n>k 0,I n J p Indeed, let y belong to the set on the right-hand side of (0.3). There are x K J p and y, [ lim (f j,x), lim (f j,x) ] such that y y, < τ 2. Choose n > k p+1 0 ( and x, I n such that I n J p and f j (x, ) y, < τ 2. Then clearly, y p+1 c j n τ,l j 2 p n + ) τ 2. p Set n m 0 =0. By (0.3), there is nm p > >n2 p >n1 p >nm p 1 such that (c 1 n τ 2 p,l1 n + τ ) [ 2 lim (f1,x), lim (f p 1,x) ] [ p, p] x K J p I n J p,n m p 1 <n n1 p and I n J p,np j 1 <n n j p (c j n τ 2 p,lj n + τ 2 p ) x K J p [ lim (fj,x), lim (f j,x) ] [ p, p] for j =2,...,m. Define ε n = τ 2 for n { } n 1 p p,...,nm p. For each n = n j p, apply [5, Lemma A.3.1] to find a nowhere dense closed set F n I n m j=1 C f j and a continuous function g n such thatg n =0outside of I n, g n 2ε n, and (f j + g n )(F n ) [ c j n ε n,l j n + ε n]. (0.4) Define F = K n N F n and g = n N g n. Note that F is nowhere dense and closed, and since ε n 0, sog is continuous. Let x K, I x be an open interval, j {1, 2,...,m} and y [lim (f j,x), lim (f j,x) ]. There exists p> y such thatx J p I. For n = n j p,theinterval I n J p and y ( c j n τ,l j 2 p n + ) ( τ 2. Then, y c j p n ε n,l j n + ε n),andby (0.4), there is t F n F with (f j + g)(t) =y. The other requirements are easy to prove. Lemma 6. Let functions f 1,f 2,...,f m DQB 1, and let ε>0 be arbitrary real number. Then, there isacontinuous function g, g ε such that f j +g ŚsB 1 for every j =1, 2,...,m. Proof. Set g 0 = 0 and F 0 =. We will procced by induction. Fix n N and assume that we have defined a closed and nowhere dense set F n 1. Define K n = F n 1 m j=1 { x R: ω (fj,x) n 1}. Then, K n is closed and 65

8 ROBERT MENKYNA nowhere dense. Use Lemma 5 to find a nowhere dense closed set F n K n and acontinuous function g n such that F n \ K n m j=1 C f j,g n = g n 1 on K n, g n g n 1 ε 2 n, and (f j + g n )(I F n \ K n ) [ lim (f j + g n 1,x), lim (f j + g n 1,x) ] (0.5) for every x K n, j =1, 2,...,m and every open interval I x. Define g =lim n g n and h j =f j +g. Then clearly, g is continuous and g ε. To prove that h j is strong wi tkowski, we will use [5, Proposition ]. Take x D hj = D fj and an open interval I x. There is n N with x K n. By (0.5), we obtain ( ) ( ) h j I Chj = hj I Cfj hj (I F n \ K n ) =(f j + g n+1 )(I F n \ K n ) [ lim (f j + g n,x), lim (f j + g n,x) ] = [ lim (h j,x), lim (h j,x) ]. (We used that g p = g n+1 on F n for p>nand g p (x) =g n (x) for p n.) So, by [5, Proposition I.3.18], h j is strong wiatkowski. Theorem 7. Let f be a Sierpi ski one function. Then, there are strong wi tkowski Sierpi ski one functions g and h such that f = g + h. Proof. If f is a Sierpi ski one function on an interval I, according to Theorem 4, there exist Darboux quasi-continuous Sierpi ski one functions f 1,f 2 defined on the interval I such that f = f 1 + f 2. Since S 1 B 1, by Lemma 6, there exists a continuous function g such that f 1 + g and f 2 + g are strong wi tkowski functions. Moreover, because S 1 + C =(lsc lsc)+c = S 1, the functions f 1 + g and f 2 + g are strong wi tkowski Sierpi ski one functions. Therefore, the function f 2 g is again strong wi tkowski Sierpi ski one function, the function f can be represented as a sum of two strong wi tkowski Sierpi ski one functions f =(f 1 + g)+(f 2 g). REFERENCES [1] BRUCKNER, A. M.: Differentiation of Real Functions, in: Lecture Notes in Math., Vol. 659, Springer, Berlin, [2] BRUCKNER, A. M.μCEDER, J. G.μKESTON, R.: Representations and approximations by Darboux functions in the first class of Baire, Rev. Roum. Math. Pures Appl. 9 (1968), [3] GUREVI, A. B.: D-continuous Sierpi ski components, Dokl. Akad. Nauk BSSR 10 (1966), (In Russian) [4] KEMPISTY, S.: Sur les functions quasicontinues, Fund. Math. 19 (1932),

9 A NOTE TO THE SIERPI SKI FIRST CLASS OF FUNCTIONS [5] MALISZEWSKI, A.: Darboux Property and Quasi-Continuity: A Uniform Approach. Wydaw. Uczelniane WSP, [6] MALISZEWSKI, A.: On the sums of Darboux upper semicontinuous quasi-continuous functions, Real Anal. Exchange 20 ( ), [7] MALISZEWSKI, A.: On the differences of upper semicontinuous quasi-continuous functions, Math. Slovaca 48 (1998), [8] MALISZEWSKI, A.: On the limits of strong wi tkowski functions, Zeszyty Nauk. Politech. ódz. Mat. 27 (1995), [9] MENKYNA, R.: On approximations of semicontinuous functions by Darboux semicontinuous functions, Real Anal. Exchange 35 ( ), [10] MENKYNA, R.: On the differences of lower semicontinuous functions, Real Anal. Exchange 41 ( ), [11] MENKYNA, R.: On representations of Baire one functions as the sum of lower and upper semicontinuous functions, Real Anal. Exchange 38 ( ), [12] MENKYNA, R.: On the sums of lower semicontinuous strong Swi tkowski functions, Real. Anal. Exchange 39 ( ), [13] MENKYNA, R.μMYDIELKA, L'.: Approximations by Darboux functions in the Baire one class, Tatra Mt. Math. Publ. 55 (2013), [14] MENKYNA, R.: Difference of two strong wi tkowski lower semicontinuous functions, Math. Slovaca 67 (2017), [15] MORAYNE, M.: Sierpi ski's hierarchy and locally Lipschitz functions, Fund. Math. 147 (1995), [16] SIERPI SKI, W.: Sur les fonctions développables en séries absolument convergentes de fonctions continues, Fund. Math. 2 (1921), Received October 13, 2016 Institute of Aurel Stodola Faculty of Electrical Engineering University of ilina ul. kpt. J. Nálepku 1390 SK Liptovsk Mikulá SLOVAKIA menkyna@lm.uniza.sk 67