COMPARISON OF SOME SUBFAMILIES OF FUNCTIONS HAVING THE BAIRE PROPERTY. 1. Introduction

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1 t m Mathematical Publications DOI: /tmmp Tatra Mt Math Publ 65 (016), COMPARISON OF SOME SUFAMILIES OF FUNCTIONS HAVING THE AIRE PROPERTY Gertruda Ivanova Aleksandra Karasińska Elżbieta Wagner-ojakowska ASTRACT We prove that the family Q of quasi-continuous functions is a strongly porous set in the space a of functions having the aire property Moreover, the family DQ of all Darboux quasi-continuous functions is a strongly porous set in the space Da of Darboux functions having the aire property It implies that each family of all functions having the A-Darboux property is strongly porous in Da if A has the ( )-property 1 Introduction In this paper we will focuse on the size of some subsets of the metric space Such studies have a long tradition, also in the case when a space is a family of functions The classical example of this kind is a result of S a n a c h obtained in 1931 Using the category method, he proved that the set of nowhere differentiable functions is residual in the space of continuous functions under the supremum metric Here, we will study the family of quasi-continuous and Darboux quasi-continuous functions in the space of functions having the aire property and the space of Darboux functions having the aire property, respectively The comparison of these families of functions will be described in porosity terms The notion of porosity in spaces of Darboux-like functions was studied among others by J Kucner, R Pawlak, Świ a t e k in [8], by H R o s e n in [13] and by G Ivanova, E Wagner-ojakowska in [6] We start with the definition of a porosity of a set at a point Let (X, d) be an arbitrary metric space, (x, r) = z X : d(x, z) <r } for x X and r>0 c 016 Mathematical Institute, Slovak Academy of Sciences 010 M a t h e m a t i c s Subject Classification: Primary 6A15; Secondary 54C08 K e y w o r d s: Darboux property, aire property, quasi-continuous functions, porous set 151 Download Date 4/7/18 7:16 AM

2 GERTRUDA IVANOVA ALEKSANDRA KARASIŃSKA ELŻIETA WAGNER-OJAKOWSKA Moreover, assume that (x, 0) = FixM X, x X and r>0 Here and subsequently, } γ(x, r, M) =sup t 0: (z,t) (x, r)\m z X Define the porosity of M at x as γ(x, r, M) p(m,x) = lim sup r 0 r + Definition 11 ([16]) The set M X is porous (strongly porous) in X if and only if p(m,x) > 0 ( p(m,x)=1 ) for each x M We finish this section with the lemma which is useful in the further part of this paper Lemma 1 ([5]) Let a, b R, a<band let P [a, b] be a set of the first category There exists a sequence of sets C n } n N, C n (a, b) for n N, with the following properties: 1 C n is closed nowhere dense set of cardinality continuum for n N; C n C m = for n, m N, n m; 3 n=1 C n P = ; 4 for each interval (e, f) (a, b), there exists a natural number n such that C n (e, f) The quasi-continuous functions Definition 1 ([7]) A function f : R R is quasi-continuous at a point x if and only if for every open neighborhood U of x and for every open neighborhood V of f (x) there exists a nonempty open set G U such that f(g) V A function f : R R is quasi-continuous if and only if it is quasi-continuous at each point For a, b R,thesymbol<a,b>stands for the interval (mina, b}, maxa, b}) Definition A function f : R R is a Darboux function if and only if for each interval (a, b) R and for each λ < f(a),f(b) > there exists a point x 0 (a, b) such that f(x 0 )=λ Let Q, D, a denote the families of all quasi-continuous functions, Darboux functions and functions having the aire property, respectively y DQ and Da we will denote the class of Darboux quasi-continuous functions and Darboux functions having the aire property 15 Download Date 4/7/18 7:16 AM

3 FUNCTIONS HAVING THE AIRE PROPERTY Let ρ be a metric in the space a given in the following way: ρ(f,g)=min 1, sup f(t) g(t) : t R }} From now on, we consider any family of functions with the above metric ρ First, we prove that the family Q is a strongly porous set in the space a of functions with the aire property For this purpose, we need some auxiliary lemmas All of them are similar to the lemmas from [5] However, in this paper, we consider other families of functions, so the proofs of these lemmas need some essential modifications For the convenience of the reader, we decided to present these proofs with details Lemma 3 Let f : R R If there exist two intervals (a, b) and (A, ) such that f 1 ((A, )) (a, b) is a nonempty set of the first category, then f Q Proof Let f : R R, (a, b) and(a, ) be such that f 1 ((A, )) (a, b) is a nonempty set of the first category Suppose on the contrary, that f Q Let x f 1 ((A, )) (a, b) As f is quasi-continuous, there exists a nonempty open set G (a, b) such that f(g) (A, ) Hence G f 1 ((A, )) (a, b), which gives a contradiction Lemma 4 There exists a function f : R [0, 1] vanishing except for some set of the first category such that 1 f is a Darboux function; f has the aire property; 3 (f,1/) Q= ; 4 there exists a function h DQsuch that ρ (h, f) =1/ Proof LetC n } n N be a sequence of the sets from Lemma 1 (for P = and [a,b]=[0,1]) For each n N, there exists a function φ n transforming C n onto [0, 1] Put φ n (x) for x C n, n N, f(x) = 0 for x R\ n=1 C n It is easy to check that f is a Darboux function having the aire property We will prove that ( f, ) 1 Q = For this purpose we will show that ( f, 1 n) 1 Q= for each natural number n> Let n> Assume that g ( f, 1 n) 1 QLetx (0, 1) be such that f(x) = 1 Hence, g(x) > 1 Asg is quasi-continuous at x, there exists a nonempty open set G (0, 1) such that g(g) ( 1, ) y the definition of f and the property of the sequence C n } n N,thereexistsapointx G such that f(x )=0 Hence, g(x ) < 1 and g(x ) ( 1, ), which gives a contradiction 153 Download Date 4/7/18 7:16 AM

4 GERTRUDA IVANOVA ALEKSANDRA KARASIŃSKA ELŻIETA WAGNER-OJAKOWSKA Now, the condition 3 is obtained immediately from the equality ( f, 1 ) ( = f, 1 1 ) n Put n=3 0 for x R\ ( 1, ) 3, 1 h(x) = for x [0, 1], linear on [ 1, 0] and [ ] 1, 3, It is easy to see that ρ (h, f) =1/ andh DQ Lemma 5 If g Q, then for each r (0, 1) and ε ( 0, 4) r there exists afunctionh a such that ε (g, r) \Q Moreover, if g DQ, then the function h is Darboux Proof Let g QFixr and ε such that 0 <ε< r 4 < 1 4 There are two possibilities: 1 The function g is constant, so g(x) =c for some c R and any x R Let f be a function from Lemma 4 Putting 154 h = g r + ε +(r ε) f, we obtain that h is Darboux and has the aire property Of course, h Q Let us show that ( h, r ε) Q= Suppose, contrary to our claim, that there exists a function φ ( h, r ε) QThus,ρ ( φ, c r + ε+ (r ε) f ) < r ε Obviously, φ c + r ε Q and ρ( φ c + r ε, (r ε) f ) < r ε, so φ c + r ( ε (r ε) f, r ) ε Q Since r ε<r<1, we have sup t R φ (t) c + r ε (r ε) f (t) } < r ε Hence, 1 (φ c + r ) ( r ε ε f, 1 ) Obviously, 1 (φ c + r ) r ε ε Q, which is impossible as from Lemma 4, ( f, ) 1 Q= Thus, ε Q= (1) Download Date 4/7/18 7:16 AM

5 FUNCTIONS HAVING THE AIRE PROPERTY Moreover, using f (R) =[0, 1], we have ɛ (g, r) () y (1) and () we obtain: ε (g, r) \Q The function g is not constant As g Q, there exists a point x such that g is continuous at x Hence, we can find an interval (a 0,b 0 ) such that x (a 0,b 0 ) and diam ( g((a 0,b 0 )) ) < ε Let Put We have A 0 = inf x (a 0,b 0 ) [ A0 + 0 J = g(x) } and 0 = sup r + ε, A x (a 0,b 0 ) ε< r ε, so (A 0, 0 ) J g(x) } + r ε ] Let C n } n N be a sequence of the sets from Lemma 1 (for P = and [a, b] =[a 0,b 0 ]) For each n N, there exists a function φ n transforming C n onto J PutC = n=1 C n and g (x) for x R\C, h(x) = φ n for x C n, n N Clearly, h has the aire property Moreover, using h (C) J and g (C) (A 0, 0 ), we obtain ρ (h, g) < r, (3) so ( h, r ɛ) (g, r) What is more, ( h, r ε) Q = Indeed, let s ( h, r ε) Suppose, contrary to our claim, that s QLetx (a 0,b 0 ) be such that h(x) =maxj Then, s(x)> A 0+ 0 The function s is quasi-continuous at x, so there exists a nonempty open set G (a 0,b 0 )withs(g) ( A 0 + 0, ) y the construction of h and the properties of the sequence C n } n N, there exists x G such that h(x )=minj Therefore, s(x ) < A 0+ 0 It means, s(x ) ( A 0+ 0, ), which gives a contradiction As s was chosen arbitrarily, we have ( h, r ε) Q= Finally, ( h, r ε) (g, r) \Q 155 Download Date 4/7/18 7:16 AM

6 GERTRUDA IVANOVA ALEKSANDRA KARASIŃSKA ELŻIETA WAGNER-OJAKOWSKA Now, we prove the second part of the lemma If g is a Darboux function, then h constructed above is also Darboux Indeed, let (c 0,d 0 ) R be such that h (c 0 ) h (d 0 )We will show that for any Z (C 0,D 0 ), where (C 0,D 0 )=<h(c 0 ),h(d 0 ) >, there exists a point z (c 0,d 0 ) such that h (z) =Z FixZ (C 0,D 0 ) If [c 0,d 0 ] (a 0,b 0 )=, thenh (x) =g (x) for x [c 0,d 0 ] Since g D, one can find a point z (c 0,d 0 ) such that Z = g (z) =h(z) If (c 0,d 0 ) (a 0,b 0 ), then put (s, t) =(c 0,d 0 ) (a 0,b 0 ) There are two possibilities: (i) Z J Lemma 1 implies that there exists k N such that C k (s, t) Since h (C k )=φ k (C k )=J, thereexistsapointz C k (c 0,d 0 )such that h (z) =Z (ii) Z/ J We consider only the case when Z (sup J, + ) andd 0 = h (c 0 ) (in other cases, the proof runs similarly, so we can omit it) It is easy to see that c 0 / (a 0,b 0 ), so c 0 <a 0 If d 0 b 0,then(s, t) =(a 0,d 0 ) As C is of the first category, there exists apointw (a 0,d 0 ) \C Obviously, h (w) =g (w) J Ash (c 0 )=g(c 0 ), we have [ sup J, h (c0 ) ) = [ sup J, g (c 0 ) ) ( g (w),g(c 0 ) ), and Z ( g (w),g(c 0 ) ) Sinceg is a Darboux function, there exists a point z (c 0,w) such that g (z) =Z WehaveZ / J and g ((a 0,b 0 )) J, so z/ (a 0,b 0 ) Thus, h (z) =g (z) =Z If d 0 > b 0,then(C 0,D 0 ) = ( h(d 0 ),h(c 0 ) ) = ( g(d 0 ),g(c 0 ) ) Hence, Z ( g(d 0 ),g(c 0 ) ) The function g is Darboux, so one can find a point z (c 0,d 0 ) such that g (z) =Z AsZ/ J and g ((a 0,b 0 )) J, z/ (a 0,b 0 ) Consequently, h (z) =g (z) =Z Theorem 6 The set Q is strongly porous in (a, ρ) Proof Let r>0 Without lose of generality, we can assume that r (0, 1) Fix g Q Lemma 5 implies that for any ε ( 0, 4) r there exists h a such that ε (g, r) \Q Thus, γ (g, r, Q) = r and 156 γ (g, r, Q) p (Q,g) = lim sup =1 r 0 r + Download Date 4/7/18 7:16 AM

7 FUNCTIONS HAVING THE AIRE PROPERTY 3 The classes of Darboux functions Theorem 31 The set DQ is strongly porous in (Da, ρ) Proof Let r>0 Without lose of generality, we can assume that r (0, 1) Fix g DQ Lemma 5 implies that for any ε (0, r 4 )thereexistsh Da such that ε (g, r) \Q Obviously, ε (g, r) \DQ, so γ (g, r, DQ) = r, and γ (g, r, DQ) p (DQ,g) = lim sup =1 r 0 r + From the last theorem, it follows that any subfamily of DQ is strongly porous in Da In particular, we have this property for the classes of functions connected with some modifications of Darboux property (see [], [3], [11]) As, according to the last theorem, DQ is strongly porous in D and each strongly porous set is nowhere dense, DQ is nowhere dense in D In 1995, A M a l i s z e w s k i [11] investigated a class of functions with a property being some modification of Darboux property He considered a family of functions with so-called strong Świ atkowski property Definition 3 ([11]) A function f : R R has the strong Świ atkowski property if and only if for each interval (a, b) R and for each λ <f(a),f(b)> there exists a point x 0 (a, b) such that f (x 0 )=λ and f is continuous at x 0 From now on, we will use the symbol D s to denote the class of functions with strong Świ atkowski property (for more details on strong Świ atkowski functions, see [11], [1], [14]) In [4], it is proved that D s DQ Thus Corollary 33 The family D s is strongly porous in (Da, ρ) The next modification of Darboux property is A-Darboux property This notion was introduced in [4] Let A P(R), where P (R) isthepowersetofr Definition 34 ([4]) We will say that f : R R is A-continuous at a point x R if and only if for each open set V R with f (x) V there exists aseta Asuch that x A and f (A) V We will say that f : R R is A-continuous if and only if f is A-continuous at each point x R 157 Download Date 4/7/18 7:16 AM

8 GERTRUDA IVANOVA ALEKSANDRA KARASIŃSKA ELŻIETA WAGNER-OJAKOWSKA Definition 35 ([4]) We will say that f : R R has the A-Darboux property if and only if for each interval (a, b) R and each λ <f(a),f(b) > there exists a point x (a, b) such that f (x) =λ and f is A-continuous at x Let D A denote the family of all functions having the A-Darboux property Let A R ThesetA is of the first category at a point x if and only if there exists an open neighborhood G of x such that A G is of the first category (see [9]) D (A) will denote the set of all points x such that A is not of the first category at x Definition 36 ([4]) We will say that the family A P(R) has (*)-property, if and only if 1 A contains the Euclidean topology, each set Ahas the aire property, 3 A D (A) for each A A It is not difficult to see that a wide class of topologies has (*)-property For example, the Euclidean topology, I-density topology, topologies constructed in [10] by E Lazarow, R A Johnson, W Wilczyński or the topology constructed by R W i e r t e l a k in [15] Also, some families of sets which are not topologies have (*)-property, for example, the family of semi-open sets On the other hand, the density topology does not have this property In [4], it is proved that if the family A has (*)-property, then D A DQ From this fact and Theorem 31, we obtain Corollary 37 The family D A is strongly porous in (Da, ρ) for each family A having (*)-property It should be added that A M a l i s z e w s k i in [11] proved that D s is dense in DQ So, we have that D s is not strongly porous in DQ Moreover, the family of all continuous functions C is strongly porous in D s (see [1]) Taking into account these facts and Theorem 31, we have Remark 38 C D s DQ Daand C is strongly porous in D s, DQ is strongly porous in Da, but D s is not strongly porous in DQ REFERENCES [1] FILIPCZAK, M IVANOVA, G WÓDKA, J: Comparison of some families of real functions in porosity terms Math Slovaca (to appear) [] GRANDE, Z: On a subclass of the family of Darboux functions, Colloq Math 117 (009), Download Date 4/7/18 7:16 AM

9 FUNCTIONS HAVING THE AIRE PROPERTY [3] IVANOVA, G: Remarks on some modification of the Darboux property, ull Soc Sci Lett Lódź Sér Rech Déform 63 (013), [4] IVANOVA, G WAGNER-OJAKOWSKA, E: On some modification of Świ atkowski property, Tatra Mt Math Publ 58 (014), [5] IVANOVA, G WAGNER-OJAKOWSKA, E: On some modification of Darboux property, Math Slovaca 66 (016), 1 10 [6] IVANOVA, G WAGNER-OJAKOWSKA, E: On some subclasses of the family of Darboux aire 1 functions, Opuscula Math 34 (014), [7] KEMPISTY, S: Sur les fonctions quasicontinues, Fund Math 19 (193), [8] KUCNER, J PAWLAK, R ŚWIA TEK, : On small subsets of the space of Darboux functions, Real Anal Exchange 5 (1999), [9] KURATOWSKI, A MOSTOWSKI, A: Set Theory with an Introduction to Descriptive Set Theory Warszawa, PWN, 1976 [10] LAZAROW, E JOHNSON, R A WILCZYŃSKI, W: Topologies related to sets having the aire property, Demonstratio Math 1 (1989), [11] MALISZEWSKI, A: On the limits of strong Świ atkowski functions, Zeszyty Nauk Politech Lódz Mat 7 (1995), [1] MALISZEWSKI, A: Darboux Property and Quasi-Continuity A Uniform Approach Habilitation Thesis, WSP, S lupsk, 1996 [13] ROSEN, H:Porosity in spaces of Darboux-like functions, Real Anal Exchange 6 (000), [14] SZCZUKA, P: Maximums of extra strong Świ atkowski functions, Math Slovaca 6 (01), [15] WIERTELAK, R: A generalization of density topology with respect to category, Real Anal Exchange 3 (006/007), [16] ZAJÍČEK, L: On σ-porous sets in abstract spaces, Abstr Appl Anal 5 (005), Received November 19, 015 Gertruda Ivanova Institute of Mathematics Academia Pomeraniensis ul Arciszewskiego d PL S lupsk POLAND gertruda@apsledupl Aleksandra Karasińska Elżbieta Wagner-ojakowska University of Lódź Faculty of Mathematics and Computer Science ul anacha PL Lódź POLAND karasia@mathunilodzpl wagner@mathunilodzpl 159 Download Date 4/7/18 7:16 AM