On Some Subclasses of DB Functions

On Some Subclasses of

VB\

Functions

Dariusz Banaszewski

1. In tro d u c tio n . Studying the behavior o f real functions H .R osen [10] defined B aire .5 functions and showed that every fu n ction B aire

.5 w ith D a rb ou x p rop erty is ąuasi-continuous. It is easy to see, that

for every fu n ction / , if / is a Baire. .5 function then / is in the first class o f Baire and there exist functions which are in the first class and not Baire .5. W e shall see below (in E xam ple 1.1) that there exists D arbou x B aire 1 fu n ction which is ąuasi-continuous but is not Baire

.5.

Let us establish som e term in ology to be used later. IR denotes the set o f all reals. A functions / : IR — > IR is said to be ąuasi-

continuous at a poin t x £ IR if for every open neighbourhoods U o f x and V o f f ( x ) there exists a n on -em pty open set W C U fi / _1(R ). A fu n ction / : IR — > IR is said to be ąuasi-continuous on IR iff it is ąuasi-continuous at each poin t x 6 IR.

A set A is said to be sem i-op en if A C IntA — [7] (b y A and IntA we denote the closure and the interior o f A ). It is well-known that a fu n ction / : IR — » IR is ąuasi-continuous iff for every n on -em p ty op en set V C IR the set / -1(E ) is sem i-open. Q denotes the fam ily o f all real ąuasi-continuous functions. M oreover we shall consider the follow ing fam ilies o f real functions defined on IR.

Isc ( u s c ) — the class o f all lower (u p p er) sem i-continuous functions,

C — the class o f all continuous functions,

V — the class o f all D arbou x functions, i.e. the class o f all fu n ction /

B i — the class o f all functions o f the first class o f Baire,

B .5 — the class o f all functions / such that for every open set U C IR

the set f ~ l {U ) is a Gs set,

■M. [M q\ — the class o f all functions / which satisfy the follow ing

con dition : if x 0 is a right (left) hand sided point o f discontinuity o f / , then f ( x o ) = 0 and there exists a seąuence {Y n}[£Li [of points o f continuity o f / ] such that f ( x n) = 0 for every n £ J\f and x n \ x 0( x n /* x 0) [3],

y — the fam ily o f all functions w ith the Y oung property, i.e. fu n c­

tions which are bilaterally dense in them selves (som e authors cali functions having this prop erty peripherally continuous),

A — the fam ily o f all almost continuous fu n ction s in the sense o f

Stallings, i.e. -functions / such that the for every open subset o f IR2 containing / contains a continuous fu n ction (n o difference betw een a function and its graph is m a d e),

Conn — the class o f all con n ectivity functions, i.e. functions / such

that f\ C is a con n ected subset o f IR2 whenever C is a con n ected subset o f IR.

It is well-know n that the follow ing inclusions hołd: M. C T>B\ and

u sc C B\ D Isc. M oreover, J. Young showed in [11] that for B aire 1 functions, D arbou x and Y oung properties are equivalent. K . Kura-

tow ski, W . Sierpiński and J. Brow n [6], [1] proved that for functions o f the first class notions o f alm ost continuity, connectednes and D arbou x p rop erty are equivalent.

T. N atkaniec showed in [9] that a function is quasi-continuous and satisfies the Y oung con dition iff for every point x 0 £ IR there exist two sequences { a J n } ^ and {2n} ^ i1 o f continuity points o f / such that

x n \ z o , z n / ' x 0 and lim n^oo f ( x n) = lim ^ o o f ( z n) = f ( x 0).

In this paper we consider the fam ily QVB\ o f all functions w hich are in the intersection o f the classes Q , V and B\. Because T>B\ = yB \ it is true that / 6 Q VB\ iff / is a Baire 1 function and f\ C (f) is bilaterally dense in / (by C ( f ) we denote the set o f points o f continuity o f / ) . In [9] T. Natkaniec proved that the follow ing inclusion holds M. C

Q-E x a m p l e 1. Let / = [0,1], C C / be a Cantor set and for each n £ J\f let J n b e the fam ily o f all com pon en ts o f the set I \ C o f the n -th order (i.e. such com pon en ts o f I \C which length is eąual to 3 _ ” ). Let A = / \ (J {J : J £ U~=i J n } and c j = infJ+2suPJ where J £ U ~ i J n • Let / vanish on A , be eąual to 1 on each o f c j , b e eąual to 1 on each o f the points m a x J , m in J and which is linear on b oth intervals [c /,m a x J ] and [m in J ,c j], T h en / is ąuasi-continuous, Baire class 1 with D arbou x property. B ut sińce the set A is not an Fa set, thus / -1(IR\ { 0 } ) is not a Gs set and conseąuently / is not Baire .5.

2. O p e ra tio n s.

Let £ b e a fixed class o f real functions. T h e maximal additive

(mul-tiplicative) class for C we define as the class off all functions / 6 C for which / + g 6 C ( f g 6 C ) whenever g € C. T h e adeąuate classes we denote by A f a( £ ) and M m{ C) , respectively. M oreover, let

A f m, „ ( £ ) = { / € £ : iffl' € £ then m in ( f , g ) € £ } ,

M m a x ( £ ) = { / e L ■ if 9 e £ then m ax ( f , g ) € £ } .

In the present paper we shall prove that

M a( Q V B i ) = £ , = M ,

M m a x { Q F B 1) = U S C Q V , M min( Q ' DB l ) = l s c Q V .

L e m m a 1 C c M a(QT>Bi).

P r o o f. Z. G randę and L. Sołtysik showed in [4] that M a( Q ) — C. M oreover A . M . B ruckner in [11] proved that A i a(T>Bi ) = C. Thus, if / £ C then / + g £ QT>Bi for each fu n ction g £ Q V B i .

Q .E .D .

L e m m a 2 M C M m( Q V B 1).

P r o o f. R . Fleissner [3] showed that M. = Mm(B>B\) and T . Natkaniec in [8] proved that M. = M . m( Q A ) . Because eąuality QDB\ = Q A B i holds, we obtain reąuired inclusion.

Q .E .D .

P r o o f. In [2] J. Farkova showed that u sc D = M .max(J)Bi) and l s c V = T . Natkaniec showed in [9] that u s c Q V = M max{Q'D) and l s c Q V = Thus m ax ( f , g ) G QT> fi VB\ = QVB\ ( m i n ( / , g ) G Q V B \ ) for every functions / G uscQ T> ( / G lscQ 'D ) and

g G Q V B i.

Q .E .D .

T h e o re m 1 C = M a{QB>B\).

P ro o f. W e need only to prove the follow ing inclusion: M a(QT>B\) C C. Let / G QVB\ \ C and let x G IR be a point o f discontinuity o f

/ . W e shall show that there exists a function g G QT>Bi such that

f + g is not a D arbou x function. W ith ou t loss o f generality we can

assume that a: is a left hand sided point o f discontinuity o f / . Because / has the Da.rboux property, the left hand sided cluster set JC~(f, x ) is a nondegenerated closed interval. Thus there exists a poin t y G

such that f ( x ) ^ y. Put

, \ S - / ( “ ) if u < x > 9\u ) = \ _{v 1 I — y it u > x.}-t

T hen g is in the first class o f Baire, is cjuasi-continuous and has the D arbou x property, but the function

f _ j 0 for u < x , | f ( x ) — y for u > x ,

has not the D arbou x property.

Q .E .D .

T h e o r e m 2 M — M m( Q' BBi ) .

P r o o f. B y L em m a 2 we need only to prove that M m( Q /B B i) C M ,

i.e. that if / G Q V B \ \ M then there exists a g G Q D B \, such that

f g QT>B\. If f M. then there exists a point x o f discontinuity o f

/ , such that either

(i) for som e unilateral neigh bou rh ood U o f x , f { u ) 0 for u G U

OT

9( u) =

Consider the first case. W e can assume that / is discontinuous from the left at x and / ( u ) / 0 for any point u E ( x — S, x) , where 6 > 0. C h oose y ± f ( x ) , y e I C ~ ( f , x ) .

Define g by

( ( f ( x - ó ) ) - 1 if u < x — 6, 9{ u) = < ( / ( w) ) - 1 if u e ( x - 6 , x ) ,

( y ~ l if u > x. It is easy to verify that g £ Q V B \. But

f f ( u ) ■ ( f { x “ £ ) ) - 1 if u < x - 6,

f g ( u ) = < 1 if u € (ar - 6 , x ) ,

[ f ( u ) • y ~ A if u > x

-Since f g { u ) — 1 for each u 6 ( x — 6 , x ) and f g { x ) / 1 , f g has not the D arbou x property.

Now suppose that / is discontinuous from the left at x and a —

f ( x ) ^ 0. Put

2a — f ( u ) if u < x ,

2a if u > x.

Evidently, g is ąuasi-continuous, Baire 1 and has the Young property. Hence g <E QT>B\. B ut observe that the function

f n ( v ) = / i/ u < x ,

^ 2a f ( a ) if u > x,

has not the D arbou x property. Indeed, we have

( f g ) i u ) = i 2a - f ( u ) ) f ( u ) = 2a f ( u ) - f 2{u) = 2a(a + f ( u ) — a) — (a -f f ( u ) — a) 2 = 2 a (a + f ( u ) — a) — (a2 + 2a ( / ( u ) — a )) — ( / ( u ) — a) 2 = 2a2 + 2 a ( f ( u ) — a) — a2 — 2 a ( f ( u ) — a) — ( / ( u ) — a) 2 = 2a2 — a2 — ( / ( u ) — a) 2 = a2 — ( / ( u ) — a) 2 for each u < x.

This shows that ( f g ) ( u ) < a2 = / ( z) 2 < /<7(-,r) and conseąuently ( f g ) cannot have the D arbou x property. Finally we obtain the reąuired eąuality

M m( Q V B i ) = u sc QT>.

T h e o r e m 3 M rnax (Q 'D B i) = u sc Q V , M m in (Q V B i) — l s c Q V . P r o o f. W e shall prove that the follow ing inclusion holds:

T h e op p osite inclusion follow s by Lem m a 3.

Let / £ QVB\ b e arbirtrary and let x b e such that f ( x ) < lim su p f ( y ) . W ith o u t loss o f generality we can assume that / is discontinuous from the left at x. Put y = | • ( m a x £ ( / , x ) + f ( x ) ) if m ax/C ( / , x ) ^ oo and y — f ( x ) + 1 otherwise. Define g by

O bviou sly g is ąuasi-continuous, B aire 1 and satisfies the Y ou n g co n ­ dition at any point u ^ x . Since f £ T>B\ and y £ so if m ax ) C ~ ( f , x ) ^ oo (if m a xA T ~ (/, x ) = o o ) then there exists a se- ąuence { x n}^L1 such that x n / * x and lim ^ o o f ( x n) = m ax ) C ~ ( f , x )

(lim ^ o o f ( x n) = f ( x ) + 2). Then

l i m ^ o o

l i m n_>00(2 y

= m ax ) C ~ ( f , x ) + f ( x ) - lim ^ o o f ( x n) = f ( x )

(

1 i

mn

—>

g[xnsj

— 2(y(x) T l ) limn_KX)

f ( x n^

^

M oreover sińce g is constant from the right at x we ob tain that g has the Y ou n g p rop erty at x.

Since m a x ( /, </) > y > f ( x ) for u < x, so m a x (/ ,g ) has not the D arbou x property. T his proves that -M TOax(Q IL t?i) = u sc QT>.

T h e inclusion l s c Q V C ■Mmin(Q'B>B i) is proved in L em m a 3. T h e o p p o site inclusion A 4 min( Q /D B i) C l s c Q V follow s im m ediately from

the facts M .max{Q'B)B i) C uscQ T>, m in ( / , ( / ) = - m a x ( - / , - j ) and / £ u s c iff —/ £ Isc. M m a x { Q V B i ) C U S C Q V .

\

3. R e m a rk s

N ow we w ould like to cali attention to som e properties o f considered classes o f functions. E xploring a m axim al m ultiplicative fam ily for

VB\ R . Fleissner defined the fam ily M . However, it turned out that

this fam ily is m ore universal, i.e. exploring other cases we get the sam e fam ily M . E xploring m axim al m ultiplicative fam ily for ąuasi- continuous functions Z. G randę in [5] defined a subfam ily M o o f M . T . N atkaniec in [9] proved that if / 6 M then for every left (righ t) hand sided poin t x o f discontinuity o f / there exists a seąuence {^ n }^ L i o f one side handed points o f continuity o f / , such that x n \ aro(arn Z aro) and f ( x n) = 0 for every n £ N . On account o f the above and our result that i) = M there arises a ąuestion w hether M o is a proper subfam ily o f M or not. T h e answer to this ąuestion is negative.

R e m a r k 1 There exists a fu n ction f £ M which vanishes only at points o f discontinuity o f f .

Let I = [ 0 ,1], C c i , J n and c j b e as in the E xam ple 1.1. T hen we define

/ ( * ) =

1 if X = C j , J e\ J n = l^ ri 0 if x E C

linear on the intervals [min J , cj ], [c j,m a x J ].

It is easy to see that / £ M , but sińce f ( x ) 7^ 0 for any continuity poin t x o f / , / ^ M

0-A n easy conseąuence o f our result is the follow ing

R e m a r k 2 F or every fu n ctio n f 6 Q VB\ the follow ing conditions are

equivalent

(1)

_{f ec,}

(2) the fu n ctio n F = ( / , g ) : IR — * IR2(F (x ) = ( f ( x ) , g ( x ) ) ) is ąuasi-

continuous, o f the first class o f B aire with D arboux prop erty f o r every fu n ctio n g £ QT>B\.

P r o o f. T h e im plication (2)= > (1) follow s from the fact that M a( Q V B 1) = C.

Indeed if / is not continuous, then there exists g £ QT>B\ such that ( / + g ) is not D arbou x. Suppose that F — ( f , g ) £ QT>Bj. Since the sum ( x , y ) i—*• x + y is a continuous function and the com p osition o f a fu n ction from the class QT>B\ with continuous fu n ction belongs (ob v iou sly ) to QT>B\, / + g £ Q V B i, to o , in con trad iction w ith the ch oice o f g.

N ow we shall verify that ( / , g ) £ Q VB\ for every fu n ction / £ C and g G Q V B i . Let / £ C and g £ Q D B\. It is easy to see that

( l i d ) G Q & i- W e shall prove that ( f , g ) £ T>. It is w ell-know n (and

easy to prove) that Conn C 'D. Thus we shall show that ( / , g ) £

C onn. Since g £ T>B\, B row n ’ s theorem im plies g £ C onn. Let A be

arbitrary con n ected subset o f IR. N otice that the fu n ction H : g\A — » IR3 defined by H ( x , g ( x ) ) = ( x, f ( x ) , g ( x ) ) for x £ A is continuous. Since g £ C on n , g\A is con n ected and therefore H( g\A) = ( f , g ) \ A is con n ected , too. Thus ( / , g) £ Conn and conseąuently ( f , g ) G Q DB\.

Q .E .D .

R eferen ces

[1] Brown J., A lm ost continuous Darbo\ix fu n ctio n s and, R e e d ’s poin t-

wise conuergence criteria , Fund. M ath. 8 6 (1974), p. 1-7.

[2] Farkova J., About the m axim um and m inim um o f D arboux fu n c ­

tions, M atem at. Ćas. 21 (1971) N o.2, p. 110-116.

[3] Fleissner R ., A note on B aire 1 Darboux fu n ction s, Real Anal. E x c h a n g e 3 (1 9 7 7 -7 8 ), p. 104-106.

[4] G randę Z., Sołtysik L., S om e rem arks on guasi-continuous real

fu n ction s, P rob . M at. 10 (1990) p. 79-86.

[5] G randę Z., On the m azim al multiplicatiue fa m ily f o r the class o f

ąuasi-continuous fu n ction s, Real Anal. Exchange, 15 (1 9 8 9 -9 0 ),

p. 437-441.

[6] K uratow ski K ., Sierpiński W ., Les fo n c tio n s de classe 1 et les

ensam bles con n exes p u n tiform es, Fund. M ath. 3 (1922), p. 3 0 3 -

[7] Levine N ., S em i-open sets and sem i-con tin u ity in topological spaces, A m er. M ath. M onthly, 70 (1963), p. 36-43.

[8] N atkaniec T ., Orwat W ., Variations on products and quotients o f

D arbouz fu n ction s, Real Anal. Exchange 15 (1 9 89 -9 0 ).

[9] N atkaniec T ., On quasi-continuous fu n ction s hauing D arbouz p rop ­

erty, in print.

[10] Rosen H., D arbouz B a ire-.5 fu n ction s, preprint.

[11] Y oung J., A theorem in the th eory o f fu n ction s o f real uariable, R end. C irc. Palerm o 24 (1907), p. 187-192.

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