ZESZYTY NAUKOWI WYŻSZEJ SZKOŁY PEDAGOGICZNEJ W BYDGOSZCZY Problemy Matematyczne 1988 z . 10
EWA STROŃSKA WSP w Bydgoszczy
ON THE FUNCTIONS APPROXIMATEIY- AND QUASI-CONTINUOUS WCHICH ABE A1M0ST EVERYWHERE DISCONTINUOUS
Denote By R the set o f r e a ls numbera.
A function f : [0,13 —► R i s sald to be ąuasi-continuous at a point x Q, i f fo r every £ > O and fo r every neigborhood U o f x q there e x is t s a nonempty open set V C U such that t f (x ) - f ( x 0^ 1 < £ f o r each point x e V.
A function f : [0,11 -> R i s said to be ąuasi-continuous i f i t l s ąuasi-continuous at each point x e C 0 ,1 l (see [1 3 ).
I t i s known there e x is t s an approximately continuous function f : [
0 , 1 3
—> R having the set o f d iscon tin u ity points of f u l i le b e s -que measure ( Zahorski [ 2 ] ) . But th is Z a h o rsk i's function i s n 't ąuasi -continuous. I s h a ll proye the follow in g theorem:Theorem 1. There e x is ts an approxim ately- and ąuasi-continuous function haying the set o f d isco n tin u ity point? o f f u l i Lebescjue measure.
P ro o f. Let ( a^ ) be seąuence o f p o sltly e r e a ls such that
[
1
)2
^ 1*- 88
-Let C1 be a Cantor's set in [0,1 J such that m (C^) » [im deno- tes the lebesąue measure ] . le t c be a Fff sets such that m ( - F^ ) ■ 0 and eyery point x e F^ i s a density point o f the set ?1.
le t ( l 1 k ) k_-| be a seąuence o f a l l components o f the set
[ o , l J -
and le t ( j ^ ) kl.i ke ® seąuence o f closed in te rv a ls such that(2 ) J1k C I 1k ( k - 1 , 2 , . . . ) •,
<3 )
m ( J1k)
^ 1
d i ; r ( } ~ ^ i ; ; 7
*
,or k - 1, *
^ ere *r
jirdenotes the boundary o f the set I 1k,and d is t / J-|k » Fr 1-]^/“
in f x 6 J1k T G Pr I 1k
There i s an approximately continuous function g1 : [ o, 1]-*R such that g ^ ( x ) - 0 f o r each point x € f o , l J - F^ and 0 <
f o r each x € F 1 ( [ 2 ] ) .
le t h^ : £0,1J -^fo,sujJ be a function such th at: oo
(
4
) h1 [ x ) ■ 0 f o r each x e [o ,lJ -U
J1k ; k*1(5 ) h^ m (R * ^ » 2 , . . « ) * and
(6 ) h1 i s continuous at each point x € [ o , l J - . I * - 7 I
continuous. The ftinction
f 1 “ *1 + h 1
i s approximately continuous and by
(
3)
, (4 ) , (5
) i t i s ą u a s i- -continuous. Let Cg C[o , 1 J
- C^ be a Cantor's set such that m ( Cg) ■ j . Denote by ( l 2k ) a eeąuence o f a l l components o f the set [ o , l 3 - Cg . There e x is ts a seąuence o f closed in te rv a ls( J2k ) k-1
8Uch
( ? )
J2k C I 2k (k - 1 , 2 , . . . ) : and
f o r k * 1 , 2, • • •There e x is t s a function approximately continuous g2 : [ 0 ,0 ~ ^ [o , a g 3 such that gg ( x )
*•
0 f o r each xę [ o , l J -
Fg and O<
gg ( x ) ^ agfo r each x € Fg ([2-J ) , where Fg C Cg i s a Fg. set such that m ^Cg - Fg ) - O and everyX£ Fg i s a density point o f Fg.
Let hg : [ 0,1 [ o , a 2J be a function such th at:
OO
hg ( x ) * O fo r each x 6 [ 0 , l 3 - U J2k * k»1
^ ^ hg ( J2k j ■
[
O,ag J ( km
1 , 2 , . . . ) ; and(
10)
hg i s continuous at each point x e [ b , l j - Cg .- 90
continuous. Then the function
f 2 * * 2 + h 2
i s approximately continuous and by ( 9 J a n d (lo ) i t i s ą u a s i-c o n ti nuous.
In g e n e ra lit y , we define fo r n - 1 , 2 , . . . , the approxim ately- and ąuasi-continuous f n : f 0,1J - ^ [ 0 , 8 ^ J , which are continuous at •ach point x € T o , l J _ c and diącontinuous at each point x e Cn ,
n-1 ^ ^
where C C [
0 , 0
U
c». i s a C antor's set o f measure .n k=1 H 2
le t us put
oo
( 11) * - 2 f n •
n-1 n
The condition ( i ) im plies the uniform convergence o f the s e r i e - ( l l j The uniform conrergence o f the s e rie ( l l j i m p l i e s the approximately continuity o f the function f at each x t f o , l J and her continuity
OQ
at each point x € r ° » 1J “
U
cn» n-1I f x £ Cn , then a l l function f n ( n / nQ and n = 1 , 2 , . . . ) o
are continuous at point x and f n i s n 't continuous and i s ąuasi -continuous at t h is point x . Hence, f i s ąuasi-continuous and i s n 't
00
continuous at each point x £ \J c .
n=1 n
We consider the space o f a l l bounded, approxim ately- and ąuasi-continuous functions f , g : r o , lJ - ^ R with Tchebyschev metric
3 ( f , g ) - sup | f ( x ) - g ( x ) | . x€fOtl J
Theorem 2. Let £ be a p o s it ir e number and le t f : [ 0 , 0 “ ^ R be a bounded approxim ately- and ąuasi-continuous function. There e x is ts a bounded approiim ately- and ąuasi-continuous function g : [ 0 , 1 J - > R haring the set o f disco n tin u ity points o f f u l i lebesąue measure and such that ę ( f , g ) ę £ .
P ro o f. I t the set D ( f ) o f a l l di&uontinuity points o f f i s o f f u l i Lebesąue measure, then g * f .
I f m ( n ( f ) ) < 1 , then we can show o f the proof o f the theorem 1 that there e x is t s au approiim ately- and ąuasi-continuous function h : [ 0 , l 3 “* r o , £ J which i s discontinuous at each point x o f a set
x € Co , l J - B. Hence the function g ■ f + h s a t i s f i e s a l l reąuired conditions.
Theorem 3. Let £ be a p o sitiv e number and l e t f j [ 0,13 ~^R be a bounded approiim ately- and ąuasi-continuous function. There e x is ts a bounded approiim ately- and ąuasi-continuous function
g : having the set o f continuity points o f p o sitiv e Lebesgue measure and such that ^ ( f , g ) ^ £ .
P ro o f. I f the set C ( f ) of continuity points o f f i s o f po- a it iv e measure, then g » f . I f m[ C ( f ) ) * 0, le t
denotes the set o f a l l continuity points o f r) - B ) » 0 and continuous at each point
- 92 -A ( f )
£
x £ [ 0 , 1 j ; th e o s c ila t io n o f f at p oin t x i s <* ) •
The s et A ^ ( f ) i s open and dense. Let I = [ a,b_] C A ^ ( f )
Z
Z
( a < h and osc f < ^ ) a c lo s e d in t e r v a l such th a t f ( a ) *•
= f ( b ) . There e x is t s such i n t e r v a l l e , beca.use monotone approxima- t e l y continuous fu n ctio n i s continuous. B e fin e
- f ( x ) + f ( a ) i f x ę 1
O i f x € [0 ,1 J - I .
h ( x )
Then th e fu n c tio n g = f + h i s a p p ro xim a tely- and ą u a s i- c o n ti nuous, g/j ; f (a ) and ( f - g |^T £ . Because C ( g ) D In t I , we have m^C (h ) ) > 0.
T h is com plete th e p r o o f. Remarąue. l e t
AQ = | f : [ 0 , l 3 " ^ R ; f i s bounded, a p p ro xim a tely- and ąu asi-con tin u ou s j ,
| f 6 AQ ; m ( C ( f ) ) = 0 j and 1 = | f € AQ ;
m(c
( f } ) > 0 J .A«o
AQ
Prom th e theorems 2 and 3 we see th a t th e s e ts AQq and AQ^ are dense both in AQ.
a l l the seta j f £ AQ ; m ( c ( f ) ) ^ j- j are open ( [
3
J , Lemma i ) , so AQ.| ł s a G j set in AQ. Every a dense G^ set i s r e s id u a l.REFERENCES
t l j S.Kempisty; Sur le s fonctions ąuaei-continues, Pund.Math. 19 (1 9 2 9 ) 184-197
[ 2 ] Z.Z ahorski; Sur l a premiere deriv ee, Trans.Amer.Math.Soc. 69 ( 1950) , pp. 1-54
f 3 j Eostyrko and S a ła t; On the structure o f some function space, Real A nalysis Exchange 10 ( 1984-85 ) , pp. 188-193
O FUNKCJACH APROKSYMATYWNIE - ORAZ QUASI-CIĄGŁYCH, KTÓRE SĄ PRAWIE WSZĘDZIE NIECIĄGŁE
Streszczenie
W tym a rty k u le pokazu ję, że w p r z e s t r z e n i fu n k c ji ograniczonych
f : [ 0 , 1 ] - » R aproksym atywnie- i qu a s i- c ią g ły c h z metryką Cze- byszewa zarówno zbi ó r fu n k c ji prawie w szęd zie n ie c ią g ły c h , ja k i