Publicacions
Ma emá iques,
Vol
35
(1991),
323-332
.
CONTINUITY
OF
THE
VISIBILITY
FUNCTION
Abs ac
ANA
FORTE
CUNTO
G
.
Bee
de ined
he
isibili y
unc ion o
a
se
S and
p o ed
i s
con inui y
in
he
in e io
o
S
.
I is
p o ed
he e
ha
he
isibili y
unc ion
o
a
plana
Jo dan
domain
S
is
con inuous
p ecisely a
he
cone
poin s
o
he
bounda y
o
S
.
1
.
No a ions
and
basic de ini ions
Unless
o he wise
s a ed,
all
he
poin s
conside ed
he e
a e
included
in
he
Euclidean
plane
E2
.
The
complemen ,
in e io ,
closu e,
bounda y
and
con ex
hull
o
a
se
S
a e
deno ed
by
C
S,
in
S, cl
S,
bd
S
and
con
S,
espec i ely
.
The
open
segmen
joining
x and
y
is
deno ed
(x
y)
.
The
subs i u ion
o
one
o
bo h
pa en heses
by
squa e ones
indica es
he
adjunc ion o
he
co esponding
endpoin s
.
The
ay
issuing
om
x and
going
h ough
y
is
deno ed
R(x
-->
y),
while
R(yx
-)
is
he
ay
issuing
om
x and
going
in
he
opposi e
di ec ion
o
ha
o
R(x
--~
y)
.
Rays
a e
always
closed
.
We
say
ha
x
sees
y
ia
S
i
[x
y]
C S
.
The
s a o
x
in
S
is
he
se
s (x,
S)
o
all
he
poin s
o
S
ha
see
x
ia
S
.
A
s a -cen e
o
S
is
a
poin
x
E
S
such
ha
s (x,
S)
=
S
.
The
con ex
ke nel
o
S
is
he
se
ke
S
o
all
he
s a -cen e s
o
S
.
S
is
s a shaped
i
ke
S
~
0
.
A
Jo dan
domain
is
a
compac
connec ed
se o
E2
whose bounda y
is
homeomo phic
o
he
uni
ci cle
.
The
open and
closed
disks
o
cen e
x and
adius
b
will
be
deno ed
U(x
;
S)
and
B(x
;
6),
espec i ely
.
I
yE
bd
S
and x E
s (y,
S)
we
say
ha
he
ay
R(x
->
y)
is
inwa d
h ough
y
i
he e
exis s
E
R(xy
-~)
such
ha
(y
)
C
in
S
.
O he wise
we
say
ha
R(x
,
y)
is
ou wa d
h ough
y
.
The
inne
s em
o
y
wi h
espec
o
S
is
he
se
ins(y,
S)
o med
byyand
all
he poin s
o
s (y,
S)
ha
issue
ou wa d
ays
h ough
y
.
A
poin
x E
S
is
a
poin
o
local
con exi y
i
he e
ex
is s
E
>
0
such
ha
S
n
B
(x,
E)
is
con ex
.
O he wise,
x
is
a
poin
o
local
noncon exi y
.
We
ema k
ha
he
dis inc ion
is
signi ican
only
o
bounda y
poin s,
since
e e y
in e io
poin
is
i ially
o
local
con exi y
.
The
se
o
all
poin s
o
local
con exi y
o
S
and
ha
o
all
poin s
o
local
noncon exi y
a e
deno ed
le
S
and
lnc S,
espec i ely
.
I is
easy
o see
ha
le
S
is
open
and
lnc
S
is
closed
in
he
ela i e
opology
o
bd
S
.
An
obs uc ion
xone
is
l
a
connec ed
componen
o lnc
S
.
32
4

A
.
FORTE
CUNTO
A
poin
x
E
bd
S
is
a
la
poin
i
x
E
le
S
n
le
CS
.
The
se
o
all
such
poin s
is
deno ed
ip
S
.
x
E
bd
S
is
an
in lec ion
poin
(and
he
se
o
all
in lec ion
poin s
is
deno ed
i p
S)
i
eí he
x
E
lnc
S
n
lnc
CS
o
x
E
lnc
S
n
le
CS
n
cl( lp
S)
.
An
a e
I'
C
bd
S
keeps
he
sense
o
cu a u e
i
ei he
I'
C
le
S
o
I'
C
lnc
S
.
I
S
is
a
closed
se
wi h
nonemp y
in e io
and x
E
S,
hen
he
se
o
c i ical
isibili y
o
x
in
S
is
he
se
c (x,
S)
=
in
S
n
bd
s (x,
S)
.
Each
poin
o
his
se
is
a
poin
o
c i ical
isibili y
o
x
in
S
.

The
poin
x
E
S
is
clea ly
isible
om
y
ia
S
i
he e
exis s
e
>
0
such
ha
B
(x,
E)
n
S
C
s (y,
S)
.
2
.
S a emen
o
he
p oblem
In
[1]
G
.
Bee
de ines
he
isibili y
unc ion
as
he
one
ha
assigns
o
each
poin
x
o
a
ixed
measu able
se
S
in
he
Euclidean
space
E
.,
he
Lebesgue
ou e
measu e
o
s (x,
S)
.
We
shall
deno e
i
(x)
.
In
[1], [2]'
and
[3]
se e al
heo ems
abou
he
con inui y
o
(x)
in
open
se s,
o in
he
in e io
o
he
se s
conside ed,
a e
demons a ed
.
The
pu pose
o
he
p esen
pape
is
o
s udy
he
beha io
o
he
isibili y
unc ion
in
he
bounda y
o a
Jo dan
domain
S
.
The
s udy
is
es ic ed
o
his
case in
o de
o
a oid
di icul ies
as
hose p esen ed
in
he
examples
o
[2]
and
[3]
.
In
his
case,
he
heo ems
shown
in
[1]
assu e he
con inui y
o
(x)
in
in
S
.
Fu he mo e,
he
bounda y
cu e
mus
ha e
ini ely
many
in lec ion
poin s
in
he
smoo h
case,
and
ini ely
many
angula
poin s
in
he
nonsmoo h
case
.
This
will
p e en
he
exis en e
o
singula
poin s
(Le
.
poin s
o
accumula ion
o
in lec ion
poin s
o
angula
poin s)
.
The
s udy
o
he
s a o
a
singula
poin
seems
almos
unmanageable
o
his
au ho
.
A
Jo dan
domain
wi hou
singula
poin s
will
be a
egula
Jo dan
domain
.
We
make
a
local
s udy
o
he
s a 's
measu e
in
a
poin
x
E
bd
S
using
he
domain
o
good
beha io
N,
ha
is
a
neighbo hood
o
x
ha ing
he
ollowing
cha ac e is ics
:
i)
i s
cen e
will
be
x,
ii)
N
includes nei he
in lec ion
poin s no
angula
poin s
o
bd
S
excep
possibly
x
i sel
.
Clea ly
ii)
assu es
ha
each
o
he
wo
suba cs
I'1
and
I'2
o
N
n
bd
S,
ha ing
x
as
one
endpoin ,
keeps
he
sense
o
cu a u e
.
We
gene alize
his
local
esul s
using
he
ac
ha
he
s a s
a e
ans
spanned
by
S
.
The
de ini ion
o
his
concep
is
gi en
below
.
Lemma
3
.1
.
Le
S
be
a
closed
se
o
he
plane,
{x
;
y
;
z}
C
S
such
ha
[x y]
U
[y z]
CS
.
Le
T=
con {x
;
y
;
z}
ha e
a
mos
one poin
w
E
lnc
S
such
ha
w
E
(x
z)
.
Then
T
C
S
.
P oo
:
A
sligh
a ia ion
in
he
p oo
o
Co olla y
2
o
[6]
yields his
lemma
ha
is,
in
he
same
spi i
o
Valen ine's
esul ,
an
use ul
consequence
o
Tie ze's
heo em
on
local
con exi y
.
Lemma
3
.2
.
Le
S
be
a
egula
Jo dan
domain
and
xo
E
bd
S
.
The e
exis s
a
domain
o
good
beha io
N
=
B(xo,
S)
.
Fu he mo e,
A=
[B(xo,
b)-
{xo}]
n
bd
S
consis s o
wo
connec ed
a cs
ending
a
xo,
such
ha
each
o
hem
keeps
he
sense
o
cu a u e
.
P oo
.
De ine
CONTINUITY
OF
THE
VISIBILITY
FUNCTION

325
3
.
Auxilia y
esul s
K
=
{z
E
bdSIz
is
an
in lec ion
poin
o
an
angula
poin }
5
=
d(x
o
,
K
-
{xo})
Rom
he
inexis en e
o
singula
poin s
i
ollows
ha
5
>
0
.
Le
F
be
he
connec ed
componen
o
[bd
S
1
B(xo,
S/2)]
ha includes
xo
.
Then
F
=
F
1
U 2
whe e
each o
hese
suba cs
ends
a
xo
and
keeps
he
sense
o
cu a u e
.
De ine
B,
=
B(xo,5/2n)
and
I'1,,
F2n
as
he
connec ed
componen s
o F1
n
B,
and
F2
n
B
espec i ely,
ha include
xo
.
Le
A,=F
n
-
[F1
nU
F2
n
]
.
Owing
o
he
simplici y
o
bd
S,
he e
exis s
a
posi i e in ege
m
such ha
B

,
10

,
=
0
.
The
ball
N
=
B

,,
sa is ies
he
hesis
.
In
he
sequel,
he
domain
o
good
beha io
wi h
espec
o
xo
will
be
deno ed
by
N
.
Le
xo
E
bd
S
and
L
a
be
a
line
h ough
xo
.
The
maximal
segmen
de e mined
by
L,,
in
S
is
he connec ed
componen
la
o
L
.
1
S
ha
includes
xo
.
The
union
o
all
hose
maximal
segmen s
is
he
an
in
xo
spanned
by
S
.
The
angula
ampli ude
o a
an
S(aaS)
will
be
he
no malizad
Lebesgue
measu e
(in
bd
N
o
he
adial
p ojec ion
o
S
om
xo
o a
bd
N
.
S
will
be
an angula
connec ed
an
i
ha
p ojec ion
is
connec ed
in
he
ela i a
opology
o
bd
N
.
Lemma
3
.3
.
ins(xo,
S)
and
s (xo,
S)
a e
ans
in
xo
.
P oo
.
Bo h
se s
a e
s a shaped
and
xo
is
a s a -cen e
o
each
o
hem
.
Lemma
3
.4
.
I
=
N
n
ins(xo,
S)
is
an angula connec ed
an
.
P oo
.
We
conside
wo
al e na i as
:
(i)
Le
u
E
I,
E
I and
xo be
no
collinea
wi h
hese
poin s
.
The e
exis
u'
E
R(uxo
->)
n
N
and
'
E
R( xo
-)
1
N
such ha
(xo
u']
n
in
S
=
0
32
6

A
.
FORTE
CUNTO
and
(xo
']
n
in
S=~
.
Assume
ha
(xo u')
and
(xo
')
a e
bo h
included
in
N
nC
S
.
De ine
/0
=
min
{d(xo,
u)
;
d(xo,
)
;
d(xp,
u')
;
d(xp,
')
}
B'
=
B(xo,
P/2)

,

Bi
=
bd
B',
ui
E
[xp
u]
n Bi

,

i
E
[xo
]
n
Bi,
u1
E
[xo u']
n
B'

,

1
E
[xo
']
n
Bi
.
Clea ly
(xo ui]
C
C
S,
and
(xo
i]
C
CS
.
Fú he mo e,
condi ion
(i)
implies
ha
i
1
L(xo
ui),
hence
[ i
ui]
n
bdS
=,A
0
.
An
analogous
a gumen
shows
ha
[ui
i]
n bd S 0
.
Le
qi
E
[ i
ui]
n
bd
S
and
pi
E
[ui
i]
n
bd
S
.
Hence
pi
=A
xo
and
qi
~
xp by
condi ion
(i),
and
bd
S
canno
c oss
nei he
(xo
ui)
no
(xo
i)
since
hese
segmen s
a e en i ely
included
in
CS
.
A
simila
a gumen
shows
ha
bd
S
canno
c oss
nei he
(xo
ui)
no
(xo i)
.
Since
N
is
he
domain
o
good
beha iou
o xp,
only
wo
suba cs
o
bd
S
(call
hem
i
and
2),
bo h
ha ing xo
as
one
ex eme,
a e
included
in
N
.
De ine
wo
ci cula
sec o s o
B'
:
such
ha
qi
E
Si
and
pi
E
S2
.
I
ollows
ha
(1)

i
C
Si,
I'2
C
S2
and
(2)
(3)
Si
=
( i
xo
ui)
and
S
2
=
(ui
xo
i)
in
(con
{ui
;
i
;
xo})
n
bd
S=
0
in
(con
{ui
;
i
;
xo})
n
bd5
=
0
I
z
E
[ui
i],
i
ollows
om
(2)
and
Lemma
3
.1
ha
[z
xo]
C
S
and
z E
s (xp,
S)
.

De ine
now
z'
E
(ui i)
n R(zxo
->)
.

Using
(3)
and
Lemma
3
.1
we
ob ain
ha
(xo
z')
C
C
S,
and z
E
ins(xo,
S)
.

A
e y
simila
a gumen
h,)lds
when
one
o
bo h
o
he
segmen s
[xo
ui],
[xo
i]
is
included
in
bdS
.
We
co-lclude
ha
I
is
con ex,
whence
i s
adial
p ojec ion
om
xo
on o
bd
N
mus
be
connec ed
in
he
ela i e
opology
o
bd
N
.
(ii)
Le
u
E
I,
E
I,
and xo
E
(u )
.
Since
R(uxo
->)
is
ou wa d,
xo E
(u
)
and
[xo
]
C
S,
i
ollows
ha
[xo
]
C
bd
S
.
The
same
a gumen
p o es
ha
[xo
u]
C
bd
S,
and
as
S
is
a
Jo dan
domain,
I
n
B'
mus
be
a
hal -ci cle
.
Clea ly,
i s
p ojec ion
om
xo
on o
bd
N
mus
be
connec ed
.
Lemma
3
.5
.
J
=
N
n
s (xo,
S)
is
an
angula
connec ed
an
.
P oo
.
We
conside
h ee
al e na i es
:
(1)
Le
u
E
J,
E
J and
u E
N
n
s ( ,
S),
while
xo
~
[u ]
.

Le
A_
con ({u
;
;
xo})
.

Then
in
A
n
bd S
=
0
since
any
c ossing
o
bd
S
o e
CONTINUITY
OF
THE
VISIBILITY
FUNCTION

32
7
[xo
u],
.
[xo,
]
o
[u ]
would

uin
he
condi iona
o
isibili
.-
I z E
[u
u]
;,
.
by
Lemma
3
.1
ollows
ha
z
E
J
.
Hence,
J
esul s
con exa
and
i s
adial
p ojec ion
om
xo
o e
bdN
mus be
connec ed
.
(2)
Le
u E
J,
E
J,
u 0
N
n
s ( ,
S),
while
xo
1
[u
]
.
De ine
b
=
min
{d(u,
x
o )
;
d( ,
xo)
}
,
B'
=
B(x
o
,
8/2),
Bi
=
bd
B',
Clea ly
u'
E
J,
'
E
J,
and
we
mayasume
ha
u'
1
N
n
s ( ',
S), since
o he wise
we
would be
in
he
si ua ion
o
pa
(1)
.
Le
z
E
(u
')
l
C
S,
p E
( '
z)
l
bd
S
l
N
l
s ( ',
S),
u
E
[xo
u]
l
Bi,
'
E
[xo
]
l
Bi
.
gE(u'z)nbdS(1N ls (ú,S)
.
The
isibili y
condi ions
assu e
ha
p
=~
xo,
q
=~
xo
and
p
=,,=
q
.
We
in end
o
p o e
ha
p
and
q
belong
o
di e en
bounda y
a es
sepa a ed
by xo
in
bd
Sn
N
.
I
may
happens
ha
p
=
'
o
q
=
u',
bu
in his
case
one
o
he
bounda y
a es
would be a
segmen ,
and
since xo
1
[p
q],
hese
poin s
mus
belong
o
di e en
a es
.
Le
us
assume
ha
p
:pÉ
'
and
q
:~
u',
and
suppose
ha
p and q
belong
o
he
same bounda y
a e
F
l
.
Wi hou
loss
o
gene ali y
assume
6,
ha
q
E
a e
(xo,
p)
C
1
.
By
he
de ini ion
o
p,
i
is
no
he
las
poin
o
1
in
B'
.
Le
A=
con ({ '
;
p
;
xo})
.
We
obse e
he
posi ion
o
I'1
wi h
espec
o
A
.
I
Pl
l
in
A=
¢,
ei he
a
double
poin
o
a
o bidden
change
in
he
sense
o
cu a u e
would
appea
in
bd
S
n
N
.
I
1
n
in
A
:~
0,
i
would
imply
a

32
8

A
.
FORTE
CUNTO
double
poin ,
a
con adic ion
o
he
isibili y
condi ions
o a
o bidden
changa
o
cu a u a
.
Hence,
p
and
q
mus
belong
o
di e en
bounda y
a cs
.
Le
Sl
=
( '
xo
u')
he
ci cula
sec o o
B'
ha
includes
z
;
S2
=
B'
n
CS
1
.
Then
bd
S
n
B'
C
S
i
.
De ine
z'
ER(zxo
-~)
n
Bi
.
Then
z'
E
J,
and
z'
E
in
S2
.
Using
pa í
(1)
we
ob ain
ha
[u' z']
U
[z'
']
C
J,
and
he
adial
p ojec ion
o
his
union
o e
bd
N
is
connec ed
.
(3)Le uEJ,VEJbu xoE[u ]
.
Le B',
Bi,
u'
and
'
be
as
in
(2)
.
Le
z
E
CS
n
B',
and
ake
p
E
[u
z]
n
s
(u',
S)
n
bd
S
q E
[ '
z]
n
s ( ',
S)
n
bd
S
Since
z,
u and
a e
no
collinea ,
i
ollows
ha
p
7~
q
.
Using
ha
z'
_
R(zxo
-)
n
Bi,
and
he
same
a gumen s
as
in
(2)
we
ob ain
[u' z']
U
[z'
']
C
J
.
Lemma
3
.6
.
Le
{x
n
I
n E
NI }
be a
sequence
o
poin s
in
S
such
ha
lim
x

,
_
xo
.
Then
ins(xo, S)
C

u n
s (xj,
S)

U
{xo}
U
Q=
~~
lim

s (x,
S)
J
U
{xo}
U
Q,
=1j=

n
',
whe e
Q
is
included
in
he
union
o a
ani e
numbe
o
maximal
segmen s
wi h
espec
o
xo
and
has
null
measu e
.
P oo
.
Le
pE
ins(xo,
S),
p

xo
.
We
conside
wo
al e na i as
:
a)
I
xo
is
clea ly
isible
om
p,
he e
exis s
a
neighbo hood
U(xo)
such
ha
u(xo)
n
S
C
s (p,
S)
.
Then
i
x,,
->
xo,
x,,,
E
u(xo)
n
Sdn
>
no
and
x
n
E
s (p,
S)
b'n
>
no
.
Hence
00

00 00
pE

n
s (x
j
,
S
)
C
un
s (xj,
s
)
.
j=no
. .

n=1 j=n
b)
I
xo
is
no
clea ly
isible
om
p,
i
ollows
om
Theo em
2
.1
o
[5]
ha
(p
xo)
n
lnc
S
:,A
0
.
Le
z
E
(p
xo) n
lnc
S
.
I
z
is
a
smoo h
poin
o
bd
S,
he e
exis s
an
obs uc ion
zone
I
C
lnc
S
ha
includes
z
.
Then
con
I' is
suppo ed
by,
a
mos ,
wo
ays
issuing
om
xo
.
Since he e
a e
ini ely
many
obs uc ion
zones,
and
o
cach
o
hem
he e
a e a
mos
wo
maximal
segmen s
o
c i ical
isibili y,
he amily
o
such
segmen s
is
ini a
.
I
z
is
no
á
smoo h
poin
o
bd
S,
a
simila
a gumen ,
based
in
he
de ini ion
o
egula
Jo dan
domain,
assu es
ha
he
amily
o
segmen s
o
c i ical isibili y
is
ini a
.
CONTINUITY
OF
THE
VISIBILITY
FUNCTION

32
9
Lemma
3
.7
.
Le
F
be
an
angula connec ed
an
in
xo
and
m(F)
i s
plana
Lebesgue
measu e
.
Then
m(F)
>
0
i
and
only
i
aa(F)
>
0
.
P oo
:
Taking
xo
as
he
o igin,
he
a ea
o
F
can
be
easily
compu ed
by
a
posi i e
adial
unc ion
(0)
ha
depends
on
he
a gumen
0,
whose
ange
o
a ia ion
is
he p ojec ion
o
F
om
xo
o e
bd
N
.
Le
m
and
n
be
he
endpoin s
o his
p ojec ion
.
I
m,
n,
and
xo
a e no
collinea ,
hen
le
A
=
con ({m
;
xo
;
n}),
and
a
be
he
angle
o med
by
R(xo
->
n)
and
R(xo
->
m)
ex e io
o
A
.
Then,
i is
clea
ha
aa(F)
=
27
-a
.
Hence,
27
(B)
in
(F)
_

l

d dB
«
o
Bo h
implica ions
o
he
hesis
ollow eadily
.
O he wise,
i
m,
n and
xo
a e in
he
same
line
L,
ake
E
F-
L,
and
A=
con ({m,
xo, n, })
.
A
e y
simila
a gumen
holds
and
(1)
is
alid
wi h
a=
7
.
Lemma
3
.8
.
Le
M
and
L
be
wo
connec ed
ans
wi h
e ex
xo,
M
CL
.
Then
m(M)
<
m(L)
i
and
only
i
aa(M)
<
aa(L)
.
P oo
.
The
se
F=
(L
-
M)
U
{xo}
is
a an in
xo
whose
p ojec ion
om
xo
o e
bd
N
is
no
necessa ily
connec ed,
bu has
a
mos
wo connec ed
componen s
.
Hence
F=
Fl
UF2
and
L
=
M
U
Fl
U
F2
.
Bo h
implica ions
o
he
hesis
can
be
ob ained
om
equali y
(1)
o
Lemma
3
.7
and
he
addi i i y
o
Lebesgue
measu e
.
Lemma
3
.9
.
m(ins(x,
S) l
N)
<
m(s (x,
S)n
N)
implies
ha
m(ins(x,
S))
<
M(s (x,
S))
.
P oo
. .
Le
us
assume
he
exis en e
o
a
sys em
o
pola
coo dina es cen e ed
a
x
and
simila
o
he
one
desc ibed
in
Lemma
3
.7
.
I
a and
/l
a e he
angula
coo dina es
o
he
endpoin s
o
he
adial
p ojec ion
o
ins(x,
S)
o e
bd
N,
and
a',
/« a e
he
co esponding
coo dina es
o
s (x,
S),
hen
aa(ins(x,
S))
=
/p
-
a, aa(s (x,
S))
=
/3'
-
a'
.
Le
e
be
he adius
o
N
.
F om
Lemma
3
.8
i
ollows
ha
a'
<
a
<
3
<
/j'
.
Using
he
no a ion
o
Lemma
3
.7
i
esul s
ha
and
a
(0)
m(s (x,
S))
=
m(s (x,
S)
1
N)
+

d dB
«~
E
0
(0)
m(ins(x,
S))
=
m(ins(x,
S)
n
N)
+
1.

d
dB
«
E
and
he
s ic
inequali y
o
he
hesis ollows
eadily
.
33
0
n-1
oo
n-oo
F om
Lemma
3
.6
i
ollows
ha
A
.
FORTE
CUNTO
4
.
The
main
heo em
and
i s
co olla ies
Theo em
4
.1
.
Le
S
be a
egula
Jo dan
domain and
xo
E
bd
S
.
Then,
he
ollowing
s a emen s
a e
equi alen
:
(i)
m(s (xo,
S))
=
m(ins(xo,
S))
(ii)
is
con inuous
in
xo
.
P oo
.
(i)
=>
(ii)
.
Assume
ha
is
discon inuous
in
xo
.
Owing
o
he
uppe
semicon inui y
o
(Bee ,
[1],
he e
mus
exis
a
sequence
{xn}
in
S
such
ha
xn
-+
xo
bu
lim
(x
n
)
<
(xo)
.
Hence,
lim
m(S (x
n
,
S))
<
m(s (xo,
S))
00 00
ins(xo,
S)
C

U
n
s (xj,
S)

U
{xo}
U
Q
n-1j-n
whe e
Q
has
null
measu e
.
F om
(2)
i
ollows
ha
00
00
m(ins(xo,
S))
<
m

U
I

I
s (xj,
S)

+
m({xo})
+
m(Q)
_
n=1j=n
00
hmoo
m
I
I

I
s (xj,
S)
I
<

hmoo
M(S (x
n
,
S))
.
n

n-
i=n
F om
his
inequali y
and
(1)
we
ob ain
a
con adic ion
o
(i)
.
(ii)
=>
(i)
.

Assume
ha
m(s (xo,
S))
=~
m(ins(xo,
S))
.

I
is
clea
ha
m(s (xo,
S))
-
m(ins(xo,
S))
>
0,
and
ha
D
=
[s (xo,
S)
-
ins(xo,
S)]
is
a
an
a
xo
ha ing
posi i e
measu e
.
F om
Lemma
3
.8
i
ollows
ha
aa(s (xo,
S))
>
m(ins(xo,
S)),
whe eas
om
Lemmms
3
.3
and
3
.4
weknow
ha
bo h
Nn
ins(xo,
S)
and
Nn
s (xo,
S)
a e
angula ly
connec ed
.
Then,
he
p ojec-
ion o
D
om
xo
o e
Ni
has
a
mos
wo
connec ed
componen s,
and
a
leas
one
o
hem
wi h
posi i e
measu e
.
Hence,
i
is
clea
ha
m(s (xo, S)
n
N)
>
m(ins(xo,
S)
n
N),
and
ha
he e
exis s
a
se
A
C
D
n
N
such ha
in
A
=,4
0
.
Selec
E
in
A
and
w
E
ins(xo,
S),
and
call
L(w,
xo)
he
line
h ough
w
and
xo
and
H+,
H
-
he
wo
open
semiplanes
in
which
his
line
di ides
E2
.
Since
V
L(w,
xo)
we
can
assume
ha
E
H+
.

The e
exis s e
>
0
such
ha B( ,
E)
CAC
D
n
N
.

Since
R(
-
xo)
is
inwa d,
he e
exis
a
poin
'
E
R( xo
->)
n
H
- n
S
n
N
wi h
( 'xo)
C
in
S
.
Clea ly
'
1
ins(xo,
S)
and
ha con adic s
(ii)
.
CONTINUITY
OF
THE
VISIBILITY
FUNCTION

33
1
is
no
Cea ly
isible
o m
'
.
Le
,,
=-
!
'
+
n-
I1
xo
.
Hence
dn
n
,
E
( '
xo)
and
n
-->
xo
.
Fa he mo e,
each
o
he
n
has
he
same
isibili y
es ic ions
as
'
wi h
espec
o
.
Le
L( ,
xo)
be
he
line
h ough
and
xo,
and
U
be
he connec ed
subse
o
S
limi ed
by
L( ,
xo)
and
no
isible
om
'
ia
S
.
Fu he mo e,
he
poin s
o
U
a e
no
isible
ia
S
om each
o
he
,
Call
S
=
S-
U
and
le
be
he
isibili y
unc ion
o
S
.
Clea ly
i
holds
dn
( n)
=
( n),
(xo)
G
V(X0)
and
om
he
uppe
semicon inui y
o
we
ob ain
lim
( n)
=
lim
( n)
<
D(X0)
G
V(X0)
n-Oo n-oo
We
say
ha
x
E
bd
S
is
a
cone
poin
i
he e
exis s
a
line
L
h ough
x
such
ha
s
(x,
S)
is
included
in
one
o
he
wo
closed
semiplanes
de e mined
by
L
.
Lemma
4
.2
.
I
S
is
a
Jo dan
domain
and
x
is
a
bounda y
poin
o
S,
he e
exis s
a
line
L
h ough
x
ha lea es
ins(x,
S)
a
one
side
o
i
.
P oo
..
As
Lmma
3
.4
s a es,
ins(x,
S)
n
N
is
an
angula
connec ed
an
.
I
such
a
line
does
no
exis ,
he e
mus
be
a
line
h ough
x
ha
in e sec s
he
adial
p ojec ion
om
xo
o
ins(x,
S)
on
bd
N
in
wo
poin s
{ ,
;
2},
whe e
a
leas
one
o
hem
(say
i)
is
no
an
endpoin
o
ha
p ojec ion
.
Hence
he e
should
exis
o
E
(xo l)
such
ha
(xo
o)
be
included
in o
in
S
.
Bu
his
should
con adic
he
ac
ha
2
E
ins(x,
S)
.
Theo em
4
.3
.
The
poin s
o
con inui y
o
he
isibili y
unc ion o
S
on
he
bounda y
o
S
a e
p ecisely
he
cone
poin s
o
ha
bounda y
.
P oo
. .
Le
x
be
a
cone
poin
o
bd
S,
and
L
a
line
h ough
x
ha
di ides
E2
in o
wo open
semiplanes
H+
and
H
-
such ha
s (x,
S)
C
cl(H+)
.
Take
u
E
s (x,
S)
-
L,
and
le
u'
E
R(ux
--~)
such
ha
(x
u']
C
H
-
.
Since
u'
~
s (x,
S),
we
ha e
wo
al e na i es
:
(a)
(x
u')
nS
=
0
.
(b)
(x
u')
n
S
:,I=
0 and
(x
u')
nCS

~
.
Each
o
his
al e na i es
p oduces
easily
a
poin
E
(x
u']
such
ha
(x
)
n
in
S
=
0,
whence
u
E
ins(x,
S)
.
We
ha e
shown
ha
s (x,
S)
-
ins(x,
S)
C
L
nS
and
he
las
se
has
null
measu e
.
Hence x
sa is ies
hypo hesis
(i)
o
Theo em
4
.1
and
is
con inuous
a
x
.
Con e sely,
assume
ha
is
con inuous
a
x
.
F om
Lemma
4
.2
he e
exis s
a
line
L
ha
p oduces
a closed
semiplane
H+
including
ins(x,
S)
.
Repea ing
a gumen s
used
in
he
second
pa
o
Theo em
4
.1
we
ob ain
ha
he
se
D
=
s (x,
S)
-
ins(x,
S)
is
a
an
ha ing one
o
wo connec ed
componen s and
has
null
measu e
.
I
ollows
ha
D
C
L,
whence
s
(x,
S)
C
H+
and x
is
a
cone
poin
.