On some properties of the spaces of almost continuous functions

UOL. 19 NO. (1996) 19-24

ON SOME PROPERTIES

OF

THE

SPACES OF

ALMOST

CONTINUOUS

FUNCTIONS*

RYSZARD JERZYPAWLAKt

InstituteofMathematics

L6d2 University

Banacha 22,90-238L6d+/-, POLAND

(Received May1, 1992andin revisedformOctober31, 1994)

ABSTRACT. Inthispaperweprove that,inthe space4of almostcontinuousfunctions(with themetric of umformconvergence),thesetoffunctionsofthe first class ofBatressuperporousateachpointofthts space

KEY WORDS AND PHRASES: Almostcontnmty; thefirstclassof Balre. blockingset.superporomty

1991AMS SUBJECTCLASSIFICATIONCODES: 54C35; 54C08.26A15.26A21

Among

manygeneralizationsof thenotionof continuity, special attention of mathematicians has recently been focused on the concept of almost continuous

(in

the sense of J. Stallings

[8])

functions. The observation of the fact that these finctions, in the case of the mapping ofclosedintervals into themselves, possess afixed point caused the investigation oftopological properties of those mappings(e.g.

[1],

[3], [5]).

Onthe otherhand, investigationsconnectedwith

algebraic operations performed on almost continuous functions were carried out (e.g.

[4], [9],

[6]).

However,

sofar, lessattentionhas been devoted tothestudyof the structure of the space of almost continuousfunctions withthe topologyof uniform convergence

(and

other topologies

defined in this set offunctions).

The present paperis anattemptatapreparatorystudyofthebasicproperties ofthis space.

Onthebasisof

J.B.

Brown’sresults includedinpaper

[1],

wemay inferthatevery Darboux

function of the first class ofBaire is an almost continuous function. Ofcourse, every almost

continuous function isaDarboux function,but neednotbeafunctionof the first class ofBaire

([1],

[7]).

Cgnsequently,

the following question arises: "how large" is the set in the space of almost continuous functions

f

I -,

I

_{with the metric of uniform confergence, composed of} functions of the first class ofBaire? Theanswertothis question is containedin Theorem1. It should be stressed here that this theorem thereby answers the question how often functions of thefirstclassofBaireoccurinthe classofDarboux functions

f

I Ipossessingafixed point. Beforeweformulatethetheoremandprovethelemmas preceding thistheorem,wegive the essential definitions andsymbolsused in this paper.

Throughout the work, we shall consider only functions

f

I

I,

where

I

[0,1].

Let

12

I

x I. The symbol0stands for thenatural netric in I

.

The bilateral closed

(bilateral

open,

etc.)

segnent with the end-points a and b is denoted by

[a, b] ((a,b), etc.).

Sinceourcosiderations arerestricted tothe unit interval, we write

[a,b]

instead

[a, b]

f31 (similarly

(a,

b)

instead of

(a,

b)f-i

I,

etc.).

Wesaythatafunction

f

X )", whereXandYare_{arbitrary topologicalspaces,}isalmost continuousif, for any openset VC X xYcontaining the graph

G(f),

V containsthegraph of

somecontinuous function g X

Thesetof all almostcontinuous fitnctions

f

I

I

andthe metricspaceconsistingof these

functions with themetric

"

ofulit’orm convergenceisdenoted bythesymbol4.

By

the letter

.A

wedenote thesetof all

f

A

which arefunctions of the first class ofBaire.

Suppose

f

X

Y. The statement that thesubset of

X

x

Y

is a blocking set of

f

in

X

x

Y

meansthatC isclosed relativeto

X

x

Y,

Ct3

(f)

q)and Cintarsects

7(g)

whenever g" X Yis acontinuousfunction. IfnopropersubsetofCis ablockingset of

f

inXx

Y,

C

is saidtobe theminimalblockingsetof

f

inX xY.

Let A

C

12.

Thenprojx(A) denotes the projection of theset

A

ontotheX-axis

and,

inthe

casewhen

A

isaclosed set,weassume

(A)r,,

{(z0, Y0)

A"

y0

min{y"

(xo,

y)

A}}.

Let

f"

I

I. If

A

C

I

(f(I)

C

B),

then by the symbol

flA

(flls)

weunderstand afunction from

A

to

I

(from

I

to

B

as asubspaceof

I).

For

0

<

(

<

<

_{1, we}defineafunction

f’

I

I

by letting:

f(x)

if

f()

[o,],

f(x)

Let

X

beametricspace. Theopen ballwith centrex and radiusr

>

0 will bedenoted by

K(z,

r).

Let

M

C

X,

x

X

and

R >

0. Then wedenoteby

7(z,R, M)

the supremum of the

set of all r

>

0for which thereexists z 5 X such that

K(z,r)

C

K(x,R) \

M. The number

p(M,x)

2-limsups_.0+

(,s,M)t

is calledtheporosity ofMat z

([10]).

M

is porous at x if

p(M,

x)

>

0. Wesay thatE C

X

is superporous at z

X

if

E

UFis porous atzwhenever

E

is porous atz

([11]).

Now we formulate three lemmas showing some properties of almost continuous functions.

These propertiesareappliedlater.

Lemma 1 Let

f:

I

--,

I

baa

function

such that

f

I)

C

[cr,/31

where

<

.

Then

f

.A

if

and only

if fllI,,al

is almostcontinuous.

Remark 1

If

f

X

--,

Y,

_where

X,

Y

_{are toplogical spaces,} is a continuous

function

such that

f(X)

C

B

C

Y,

then

flls

is a continuousfunction, too.

However

it is not

diJflcult

to construct

an example

of

atopology

T

defined

in

I

and analmost continuous

function

I

(1, 7r)

such that

9(I)=

[1/2,1]

and_gilt1/2,1] /s not almostcontinuous.

Lemma

2

([6])

A

function

f:

I

I

isalmostcontinuous

if

andonly

if flt,bl

isalmost contin-uous

for

anya, b

[0,1].

Remark2 It is known that there exists an almost continuous real

function defined

on some compactsubset

K

of

theplane, such that therestriction

of

this

function

to the set

Int

K

isnot almost

continuous([6]).

Lemma

3

Let

f:

I

[,

3]

bea

function

for

which there existsaninterval

(a, b)

C1 such that

fll\(a,li

and

fll\t0,b)

avealmostcontinuousand

f

isnotalmostcontinuous. Then

projx(h’l)fq(a,

b)

Proof. Itissuiticicnt to prove(see,forexavnple,

[3],

Theorem 1;

[9],

Lemma4;

[6],

Theorem 1.1.2) thatprojx(Kl)f3 (a.b)

.

Assume,

tothecontrary, that

prodx(Kl)

(o,b)

0.

Thus,

according tothe above-cited theor(’ns, either

I(

C

{(x,y)

x

a}

or

lf

C

{(x,y)

x

b}.

Without lossofgeneralityassume t}at

Kl

C

{(x.y)

x

a}.

Ofcorse, K_f

G(f)

0,and so,

Kl9(f*)

wherc

f"

.flt(,,].

L(’t us considcrtheset A

((I

(a.

1])

x

[o,])

KI.

Thus

Aisaneighbourhoodof(.f’)(in (1 (a,

1])

x

[&,/]).

Thismeans that Acow,talusthegraphof somecontinuous functiong*:

I

(a,l]

[a, fl].

Let

(for

x G

I)

9*(x)

if xEl\(a,l],

g*(a)

f

x

>_

a.

Then g is continuousand

K

_Igl

(g)

q), which isimpossible because

K!

is theblockingsetof

f.

The contradictions obtained ends theproofofthelemma.

Theorem 1 The set

4

isperfectand superporous at each point

of

the space

.A.

Proof. AccordingtoJ.B. Brown’s theorem and the well-known fact

(see,

for example,

[2],

Theorem

2.3.4])

that the limit ofa uniformly convergent sequence ofDarbouxfunctions of the

firstclass ofBaire isa Darbouxfunctionof thefirst class ofBaire, wededuce that

A1

isclosed

in ,4.

Weshall showthat

A1

isdenseinitself.

Let

f

E

.A,

> 0and letxo

(0,1)

beacontinuity point of

f.

Then either

f(xo) <

or

f(xo) >

0. Withoutlossofgeneralityassumethat

f(xo) <

1. Letnow0

<

#1

<

3abea number such that

f(Xo)

+

2-gl

<

1. Then thereexists(f

>

0 such that

[z0

6,x0

+

6]

C

(0,

1)

and

f([xo

(5,Xo

+

6])

C

(f(xo)

#,,f(xo)

+

#,).

Let us consider g

I

I

defined by letting:

g(x)

0 for

x

q

[xo-6, xo+6],

g(xo)

#1 and g is linear in

[xo-6,Xo]

and

[xo,

xo+6].

Thus g is

continuous,andso

(see

[2],

Theorem

2.3.2),

h

f

4- g ,41. Itis not difficult toseethat

f

h

and

*(f,h)

<

#.

Inthis way the factthat

.A

isperfecthas been demonstrated.

Now,

weshall show that

41

is superporous ateach point of thespace

.A.

So,

let

M

beanarbitraryporousset at t. Assumethenotationfrom the definition ofaporous

set. Let

p(M,t)

2a

>

0. Thismeans that thereexistsasequence

{R,}

such that

P

"

0and

lim

7(t,

R,,,

M)

a.

Letnbeafixednumber. Fromthedefinitionof

7(t, R,,

M)

itfollows that

(for

nsufficiently

large)

thereared

A

and

,

>_

7(t,R,,M)-

.

R,, >

0, such that

K(d,,)

C

K(t,t)

\

M.

Moreover,

if

K(d,

5_)

;3

A,

O,

then

7(t,R,,,M

U

A,) >

5-

(2)

Letusconsiderthecasewhenthereexists

o

K(d,

)

f’l

4.

Weinfer that

K(2,

--)

C

K(d,-)

C

K(t,P).

(3)

belongingto(a,b) wedefinetlwfollowing equivalewerelation:

a"

.

y

f

aTdonly

i.f

x y

where denotes thesetof alrationaltmmbers.

Let denote the set of all e(luivalenceclasses of the above relation. Foreach x

(a,b),

let

B

fi beaset such that x

B.

By

the symbol wedenote an arbitrary mapping from

onto

B1

where

B

stands for t]e set of all functions of the first class of Baire from

I

to

[(x0)-

,

(x0)+

]].

Moreover,

wedefinethe function9 by letting

(x)

if

xaorxb,

g(x)

((B))(x)

zf

z

e

(a,b).

Weshall showthat

eA.

(4)

Suppose

on the contrary that 9

it

A.

Then, according to Lemma 2, we can show that the

function91

I

I

defined by:

(a)if

x<_a,

(,.)

() iI

(,),

p(b)

if

x>b does notbelong to

From Lemma itfollows thatg*

gll[,(o)-,.(}+l

isnotalmost continuous, and so,there

exists a minimal blocking set

Ka.

C Ix

[V(x0)

i, T(x0)

+

i]

ofg*.

By

Lemma2wededucethattheassumption ofLemma3

arefulfils and so

(a, b)

Ka.

containssomenondegenerateinterval

[c,

d].

Thus the set U

{(x,min{y

I’(c,d)

K,.})’x

<_ c}U

U{(x,

rnin{y

I (d,y)

Ka.})

x

>_ d}U

u(ga.

13

{(x,y)

c

<_

x

<_

isthegraphofsome_{function r/}

B.

Let

B,

B

such that

(B,)

r/and

z

B,

FI

[c,

at].

Then

g*(z)

g(z)

r/(z)

and,consequently,

(z,g*(z))

K.,

which contradicts the assumption that

K.

istheblockingset of g*; thuscondition

(4)

issatisfied. Weinfer that:

However,

bycondition

(5),

wehave

K(g,-)

CK(cp,

e).

(6)

Now,

we shall show

that,

for each function( 4,, we have

K(g,

).

Assume

to the contrary that thereexists such that

K(g,

)

3

.A.

Therefore,

in theinterval

(a, b)

there

lies some continiuty point yoof

.

We

deducethat

(yo)

[(xo)-

,

(xo)+

].

Without

loss ofgenerality assume, that

(y0)

[q(z0),q(x0)+

]

(if

(y0)

[(x0)-

,

(xo)],

then the proofis

analogous).

Thus thereexists a nondegenerate interval

(p,

q)

C

(a, b)

such that

-I0"

Alllyig (9).

,,

0,, (for ,lliciely large). Thus, to

the lrool,il ,llici’s Io ol,rvelhat"

tSupported

IL

KBNresearcll grantI’B691/2/91.

References

[1]

BrownJ.B., Almostcontinuou.Darboux]’unctionsandReed’s pointwise coneeyencccriteria,

Fund. Math. 86 (197-1)1)p.2’19-263.

[2]

Bruckner

A.M.,

Differ,nliatio,,

of

re.alfunction.s,Springer- -Verlag

(1978).

[3]

Kellum

K.R..

,.tlmo.,t conltn,ous

functions

on,

I",

Fund. Math. 79

(1973),

1)p.213-215.

[4]

Kellum

K.R.,

Sumsandhmt,

of

almost continuousfunctions,Coll. Math.31 (1974),

pp.125-128.

[5]

Lipifiski J.S., On a problem concernin9 the almost continuity, Zesz. Nauk. U.G. 4

(1979),pp.61-6,1.

[6]

Natkaniec

T.,

Almostcontinuity,Bydgoszcz

(1992),

pp.l-132.

[7]

Reed

C.S.,

Pointwiselimits

of

sequences

of

functions, Fund.Math. 67

(1970),

pp.183-193.

[8]

Stallings

J.,

Fixedpoint theorems

for

connectivitymaps, Fund. Math. 47

(1959),

pp.249-263.

[9]

Stronska

E.,

Algebraicstructuresgenerated by

Ta

quasi-continuous and almostcontinuous

functions

on

R

,

Real Anal. Exch. 16

(1990-91),

pp.169-176.

[10]

ZajSek

L.,

Sets

of

a- porosity and sets

of

a-porosity(q),

(asopis

P&st. Mat. 101

(1976)

pp.350-359.