• Nie Znaleziono Wyników

On the functions approximately- and quasi-continuous which are almost everywhere discontinuous

N/A
N/A
Protected

Academic year: 2021

Share "On the functions approximately- and quasi-continuous which are almost everywhere discontinuous"

Copied!
7
0
0

Pełen tekst

(1)

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*

(2)

- 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

jir

denotes 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

(3)

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 x

ę [ o , l J -

Fg and O

<

gg ( x ) ^ ag

fo 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 ( k

m

1 , 2 , . . . ) ; and

(

10

)

hg i s continuous at each point x e [ b , l j - Cg .

(4)

- 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-1

I 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

(5)

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

(6)

- 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.

(7)

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

Cytaty

Powiązane dokumenty

Abstract. In this note we present the definition, examples and some properties of quasi N–almost periodic functions, i. certain almost periodic functions in the sense of Levitan...

Pierwsza część ma na celu zbadanie potencjalnego wpływu nauczania społecznego prymasa Augusta Hlon- da na kształtowanie się metody pracy duszpasterskiej kard.. Część druga

E ng elk ing, G eneral Topology, Po lish

Białystok 2009.. Biblioteka Niepaństwowej Wyższej Szkoły Pedagogicznej w Białymstoku, zwana dalej Biblioteką jest ogólnouczelnianą jednostką organizacyjną o

4. Nie jestem kandydatem na żołnierza zawodowego lub żołnierzem zawodowym, który podjął studia na podstawie skierowania przez właściwy organ wojskowy lub otrzymał

It is shown that the existence of continuous functions on the interval [0, 1] that are nowhere differentiable can be deduced from the Baire category theorem.. This approach also

ZESZYTY NAUKOWE WYŻSZEJ SZKOŁY PEDAGOGICZNEJ w BYDGOSZCZY Problemy Matematyczne 1985 z... X there exists an open neighbourhood which satisfies the second

In order to validate the co-simulation, a reference monolithic simulation was conducted in PowerFactory, too, in which standard dynamic models and dedicated DSL for the