ON
APPROXIMATE
SEMI-CONTINUOUS
A,
Iti
this papsr approximately
and
tbo
ced.
c o n c e p t s of
of
tions i n r e g a r d
closed and
open
approximately
been established and the
behaviour
to m e t r i c a l l y
of
closed
have been
sets h a v e
semi-continuous
approximately
sets a n d
been
1 . I n t r o d u c t i o n 3 L e t f(x)
upper
measurable
be a b o u n d e d
functions
f(x)
metrically
open
if
lozuer
a p o i n t of density such
that
/<*)>/(?)-«.
^
i s a. u. s. c. { o r a. I. s. c.)
It
x=
But
is
%, then
obvious
t h a t , i f f(x)
there
there exist f u n c t i o n s
a
or
in
[a, b] w i t h % as
SC.
at each \ of
t h e n i t is
[a, b],
is
approximately
continuous
* — 5
upper
a
said
to
{ [ J , p . 182}
n
(
at
0
if
,v i s
— ,i
jf
—
=
for a l l other rationals.
Here
59
3
l ^
a
s o
conversely.
semi-continuous
The f o l l o w i n g is an i l l u s t r a t i o n :
1
exists
b].
w h i c h are n e i t h e r
x
in-
approxi-
that
semi-continuous
set
a t a p o i n t £ w h e r e i t m a y b e a. u. s.
f[ )~
the
*£S.
i t i s b o t h a. I. s. c. a n d a. u. s. c. a t
proximately continuous
is
s.
x e
approximately
in
t h a t f(x)
a. u- s. c. a t x = % i f
i f there exists a measurable
I f f(x)
have
real function defined
s h o r t a. I. s. c. a t x ~\
be a. u. s. c. ( o r a. I. s. c.) i n [a,
func-
sets
L
is s a i d t o be
have
semi-continuous
[fli b] w i t h | as a. p o i n t o f d e n s i t y [ ] .such
/(*) < /(i) + « ,
Similarly,
introdu-
studied.
or i n s h o r t
semi-continuous
s e t SC.
defined
been
t e r v a l [a, b]. L e t | £ [a, b] a n d £ > 0 be a r b i t r a r y . W e s a y
mated
SETS
BAISNAB
semi-continuous functions
metrically
Some p r o p e r t i e s
P.
FUNCTIONS AND A L L I E D
irrational
l
c.
nor
ap-
60
A.
The purpose
P.
BAISNAB
of t h e present paper is to prove
mately semi-continuous
functions.
To do
some properties of a p p r o x i -
t h i s , wc are
led to
c o n c e p t o f m e t r i c a l l y c l o s e d s e t s a n d as a c o n s e q u e n c e
to w h a t extent a p p r o x i m a t e l y c o n t i n u o u s
t h e a i d o f t h e sets c o m p l e m e n t a r y
Throughout
we
the paper, we s h a l l
f u n c t i o n s c a n he c h a r a c t e r i s e d
-f- g(x)
Suppose
f(x)
is also a. I. s. c. at
Proof :
and C
Let A
such
always
consider
those
and
/ ( * „ ) > B,
g(x)
L e t S = S-L C\ S .
Then
2
w i t h / ( J K ) ~\~ g(x„)
x
at
are
x£
arc
single-va-
[a, b].
0
Since
C > A.
and *
Then
> A.
0
g(x^) > C, B +
S is measurable
+ g(x) > B - f C > A f o r
Then
f(x)
and S
a n d g(x)
> C f o r x (z
is a p o i n t o f
0
arc.
g(x)
sets S
t
t
there exist B
and
with
2
*
0
S.
t
d e n s i t y of S a n d
€ S. S i n c e ¿4 i s a n y n u m b e r w i t h A < f{x )
-f- #{.v )
0
0
follows.
Z.e£ an increasing
T h e o r e m 2.
sequence
suppose
Lt f (x)
that all the functions
is a. I. s. c. at
n
of
functions
^ / „ W ^
fM^Ux)^
be given;
which
0
as p o i n t o f d e n s i t y a n d s u c h t h a t / { * ) > B f o r x £ S
f(x)=
which
a. I. s- c
are
by d e f i n i t i o n there exist measurable
0>
the theorem
sets
x.
be a n y n u m b e r
thaL
a. I. s. c. a t x
/(*)
with
a n d defined i n [a, &].
2. T h e o r e m 1.
f{x)
the
investigated
to m e t r i c a l l y closed sets.
s u b s e t s o f t h e s e g m e n t f a , b] a n d t h o s e r e a l f u n c t i o n s
lued, bounded
introduce
have
are a . /. s. c . at x
f {x)
n
£ [a, b].
Then
Then for a l l
suffi-
u
x.
a
/I-+CO
Proof:
Let A
be a r e a l n u m b e r
ciently large values
n ' s , say
s u c h t h a t f(x )>A.
v
o f n, f„(x )>A'
We choose oue o f
(i
m a n d keep i t f i x e d . Since f
t h a t there exists a measurable
s u c h t h a t / , „ (x) > A
if
m
set S C
the
(x) i s a. I. s. c. a t x = x
¿.1 " w i t h x
a
x £ S. Since, f o r e v e r y
f(x)
x >
as p o i n t o f
m
sum of
the series
Definition:
t o b e metrically
T h e o r e m 4.
metrically
closed.
that all the terms
converges
in [a, b]. If
is also a. I. s. c. at
A point «
every neighbourhood
//
of
n
i ' u„ (*) ore
is a. I. s. c. at x,„
and
>
A
nonnegathen
any set,
the
x.
0
i s s a i d t o be a m e t r i c l i m i t p o i n t o f a set S,
i f i t contains
A is
the series
each u (x)
o f a, t h e r e i s a p o i n t o f
closed
density
1 a n d 2 we get t h e f o l l o w i n g :
Suppose
T h e o r e m 3.
these
i t follows
follows.
0
ti-ve and the series
0
of
(x) w e h a v e f(x)
^ f
i f x £ J . A s A is any n u m b e r w i t h A < /(-v ), the theorem
Combining theorems
values
d e n s i t y o f S.
a l l its metric limit
then
the set
of
A set S
is
if in
said
points.
metric
limit
points
of
A
is
ON A P P R O X I M A T E
Lot A
Proof:
v
SEMI-CONTINUOUS
FUNCTIONS AND A L L I E D
d e n o t e t h e set o f m e t r i c l i m i t p o i n t s of
m e t r i c l i m i t , p o i n t o f A¡_. W e a r e t o s h o w t h a t % £ A .
sity of A
and consequently
l
bourhood
is
A
a n d l e t i be a
L e t 1% be a
t
h o o d o f %. S i n c e % i s a m e t r i c l i m i t p o i n t o f A,,
61
SETS
there exists a
neighbour-
point of
den-
a p o i n t o f d e n s i t y A i n 1%. S i n c e / § i s a n y n e i g h -
o f %, % i s a m e t r i c l i m i t p o i n t o f A. H e n c e % £ A±
and
the
theorem
proved.
T h e o r e m 5.
//
for
any real number
E[x£
is metrically
closed,
Let e > 0
Proof;
[a,
then f(x)
a the
[a, b];
set
/OO^oc}
is a. « . s. c. in [a,
be a r b i t r a r y a n d ^ be
b].
taken at random on the segment
b].
Let
S^Eixtia,
b);
f(x)^f&)
+ s)
=
b}:
f{x)<f(l)
+
and
So, 5
U ^
L
-
[a, b] a n d S
t
sed a n d I £ S ,
belong
to S.
C\ S — <Z>. S i n c e Si i s , b y h y p o t h e s i s , m e t r i c a l l y c l o 2
there exists a neighbourhood
3
sity and such
which
^et S w i t h % as a p o i n t o f
den-
that
if
f(x)<f(l)-\-s
H e n c e f{x)
of % almost a l l points of
Hence there exists a measurable
s
s).
X
£ S .
i s a. u. s. c. a t x — %• S i n c e | i s a n y p o i n t i n
[a, b]
the
theorem
follows.
Note;
I t is o b v i o u s t h a t theorems 1, 2 a n d 5 have analagous
i f o n e r e p l a c e s a. I. s. c. b y a. u. s. c. a n d
3.
I f S i s a n y m e t r i c a l l y c l o s e d set i n [a, b] a n d £ £ S.
t h a t there exists a neighbourhood
the complement
A p o i n t | o f a set
i f there exists a neighbourhood
o f S.
of % almost a l l points
S is s a i d t o be a
metric
interior
of % almost a l l points of w h i c h
A set S i s s a i d t o b e metrically
terior p o i n t of
then
of w h i c h
it
follows
belong
to
o f S- T h i s c o n s i d e r a t i o n s u g g e s t s t h e f o l l o w i n g d e f i n i t i o n .
Definition i
S,
counterparts
viceversa.
point
are
of
points
open i f e v e r y p o i n t o f S i s a m e t r i c i n -
S.
The following results follow immediately :
(1) I f A i s a m e t r i c a l l y
closed set,
then
its
complement
is
metrically
open.
(2) I f A i s a m e t r i c a l l y o p e n s e t , t h e n i t s c o m p l e m e n t
is m e t r i c a l l y closed.
62
A.
P . BAISNAB
(3) I f A a n d B a r e m e t r i c a l l y o p e n s e t s , t h e n A U
r i c a l l y open
(4) U n i o n o f
opeu
an a r b i t r a r y f a m i l y of m e t r i c a l l y open
4.
I t m a y be of
some interest to
ask
h a v e attempted to e s t a b l i s h some of
introduce another
t y p e o f set clefiued
what
relations
T h e o r e m 6.
metrically
open
these
these
Let
metrically
connections.
To
do t h i s ,
we
below.
p o i n t of S
is a
3be a boundsd
f(x)
set 0,f~~ (0)
function
is metricolly
l
open,
defined
[a, & ] .
in
then f(x)
//
every
for
is approximately
conti-
in [a, b\.
Conversely,
i f f(x)
is approximately continuous
and /
_
fulfils
1
d i t i o n (N)
{ ['¿1, p . 2 2 4 } i n [a, b] Lhen f o r a n y m e t r i c a l l y o p e n set O,
an .4-type
set.
Proof •
set w i t h
Let | £
span less
hypothesis,
/
(O) is m e t r i c a l l y
— 1
f i r s t p a r t of the theorem
Then % £ £~MO).
there
l
and
at * =
| f(x)
— f(%)
Since % is any
\ < s i f x £ S.
be
— 1
approximately continuous
in
So,
the
f a , b]. L e t O he a
(O) is v o i d , the t h e o r e m is p r o v e d .
is a p p r o x i m a t e l y c o n t i n u o u s
S o , l e t % £ [a, b]
the condition
(TV) i n [a, b], t h e r e e x i s t s a set A% w i t h £ as a p o i n t o f
C O. L e t t h e p o i n t % be
by
by
measu-
p o i n t i n [a, b],
S i n c e f(x)
set d e n o t e d
adjoined to the
at x = % a n d
set
A\
and
fulfils
density
the resulting
A\.
Then
I € ^
C
/"MO).
So,
/-MO)
Hence every
is a n ^4—type
j.
(O) i s
Since,
exists a
a n d fC%) t O.
a n d f{At)
1
follows.
l e t f(x)
m e t r i c a l l y open set. I f /
/(|).
o p e n a n d % £ f~ (0)
of d e n s i t y
is a p p r o x i m a t e l y c o n t i n u o u s
Conversely,
the conf"'
[a, b] a n d n > 0 be a r b i t r a r y . ' L e t Q be a m e t r i c a l l y o p e n
than s and containing
r a b l e s e t S w i t h \ as a p o i n t
f(x)
metrically
functions. I n the following
A set $ i s s a i d t o b e a n / 1 - t y p e s e t i f e v e r y
Definition 3
point of density of
nuous
sets is a
set.
open sets bear w i t h t h e a p p r o x i m a t e l y c o n t i n u o u s
we
B a n d A C\ B a r e m e t -
sets.
p o i n t of
:
U
A\.
/ ^ ' ( O ) is a p o i n t of d e n s i t y of / ~ ( 0 ) ,
set.
T h i s completes the proof.
1
i.e.
/~MO)
[:
ON
APPROXIMATE
SEMI-CONTINUOUS
FUNCTIONS
ANDALLIED
63
SETS
REFERENCES
[ ] JBPPEREY, R . L . :
T h e Theory
[ ]
Theory
L
2
SAKS, S .
:
DEPARTMENT
UNIVERSITY
WEST
OF PURE
of F u n c t i o n s of a r e a l v a r i a b l e , TORONTO-PRESS, 1 1 8 , ( 1 9 5 3 ) .
of t h e I n t e g r a l , N o w Y o r k ,
HAFNER, ( 1 9 3 7 ) .
(Manuscript
MATHEMATICS,
received
August
17,
OP BURDWAN,
BENGAL,
INDIA.
Ö Z E T
Bu
araştırmada
bunun
ithal
(yaklaşık
i n c e l e n m e s i için
edilmiştir.
edilmiş
ve
Yaklaşık
yarı-surekii
(metrik
fonksiyon)
olarak kapalı v e y a
yarı-Hürekli
b u çeşit fonksiyonların
tanımı o r t a y a
açık
fonksiyonların
metrik
olarak
açık cümlelerle i l g i l i özelikleri
cümle)
bazı
atılmış
özelikleri
kapalı ve m e t r i k
incelenmiştir.
ve
kavramları
ispat
olarak
1963)