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4. CONTINUITY

27

1

.

f

is monotone and

f

C(

A, B

) iff

f

C(

A, B

)

C(

ι

A

|

R

|

, ι

B

|

R

|

)

iff

f

C(

A, B

)

C(

ι

A

|

R

|

>

, ι

B

|

R

|

) iff

f

C(

ι

A

|

R

|

, ι

B

|

R

|

)

C(

ι

A

|

R

|

, ι

B

|

R

|

).

2

.

f

is strictly monotone and

C(

A, B

) iff

f

C(

A, B

)

C(

ι

A

|

R

|

>

, ι

B

|

R

|

>

)

iff

f

C(

ι

A

|

R

|

>

, ι

B

|

R

|

>

)

C(

ι

A

|

R

|

<

, ι

B

|

R

|

<

).

FiXme

: Generalize for arbitrary posets.

FiXme

: Generalize for

f

being a funcoid.

Proof.

Because

f

is continuous, we have

h

f

ι

A

|

R

|i

{

x

} v h

ι

B

|

R

| ◦

f

i

{

x

}

that is

h

f

i

∆(

x

)

v

∆(

f

(

x

)) for every

x

.

If

f

is

monotone,

we

have

h

f

i

(

x

)

v

[

f

(

x

);

[.

Thus

h

f

i

(

x

)

v

(

f

(

x

)), that is

h

f

ι

A

|

R

|

i

{

x

} v h

ι

B

|

R

|

f

i

{

x

}

, thus

f

C(

ι

A

|

R

|

, ι

B

|

R

|

).

If

f

is

strictly

monotone,

we

have

h

f

i

>

(

x

)

v

]

f

(

x

);

[.

Thus

h

f

i

>

(

x

)

v

>

(

f

(

x

)), that is

h

f

ι

A

|

R

|

>

i

{

x

} v h

ι

B

|

R

|

>

f

i

{

x

}

, thus

f

C(

ι

A

|

R

|

>

, ι

B

|

R

|

>

).

Let now

f

C(

ι

A

|

R

|

, ι

B

|

R

|

).

Take any

a

A

and let

c

=

n

b

B

b

a,

x

[

a

;

b

[:

f

(

x

)

f

(

a

)

o

. It’s enough to prove that

c

is the right endpoint (finite or infinite) of

A

.

Indeed by continuity

f

(

a

)

f

(

c

) and if

c

is not already the right endpoint

of

A

, then there is

b

0

> c

such that

x

[

c

;

b

0

[:

f

(

x

)

f

(

c

). So we have

x

[

a

;

b

0

[:

f

(

x

)

f

(

c

) what contradicts to the above.

So

f

is monotone on the entire

A

.

f

C(

ι

A

|

R

|

, ι

B

|

R

|

)

f

C(

ι

A

|

R

|

>

, ι

B

|

R

|

) is obvious. Reversely

f

C(

ι

A

|

R

|

>

, ι

B

|

R

|

)

f

ι

A

|

R

|

>

v

ι

B

|

R

|

f

⇔ ∀

x

R

:

h

f

ih

ι

A

|

R

|

>

i

{

x

} v

h

ι

B

|

R

|

i

h

f

i

{

x

} ⇔ ∀

x

R

:

h

f

i

>

(

x

)

v

f

(

x

)

⇔ ∀

x

R

:

h

f

i

>

(

x

)

t

{

f

(

x

)

} v

f

(

x

)

⇔ ∀

x

R

:

h

f

i

>

(

x

)

t {

x

} v

f

(

x

)

⇔ ∀

x

R

:

h

f

i

(

x

)

v

f

(

x

)

⇔ ∀

x

R

:

h

f

ih

ι

A

|

R

|

i

{

x

} v h

ι

B

|

R

|

i

h

f

i

{

x

} ⇔ ∀

x

R

:

f

ι

A

|

R

|

v

ι

B

|

R

|

f

f

C(

ι

A

|

R

|

, ι

B

|

R

|

).

Let

f

C(

ι

A

|

R

|

>

, ι

B

|

R

|

>

). Then

f

C(

ι

A

|

R

|

>

, ι

B

|

R

|

) and thus it is mono-

tone. We need to prove that

f

is strictly monotone. Suppose the contrary. Then

there is a nonempty interval [

p

;

q

]

A

such that

f

is constant on this interval. But

this is impossible because

f

C(

ι

A

|

R

|

>

, ι

B

|

R

|

>

).

Prove that

f

C(

ι

A

|

R

|

, ι

B

|

R

|

)

C(

ι

A

|

R

|

, ι

B

|

R

|

) implies

f

C(

A, B

).

Really, it implies

h

f

i

(

x

)

v

(

f x

) and

h

f

i

(

x

)

v

(

f x

) thus

h

f

i

∆(

x

) =

h

f

i

(∆

(

x

)

t {

x

} t

(

x

))

f

(

x

)

t {

f

(

x

)

} t

f

(

x

) = ∆(

f

(

x

)).

Prove that

f

C(

ι

A

|

R

|

>

, ι

B

|

R

|

>

)

C(

ι

A

|

R

|

<

, ι

B

|

R

|

<

)

f

C(

A, B

). Really, it

implies

h

f

i

<

(

x

)

v

<

(

f x

) and

h

f

i

>

(

x

)

v

>

(

f x

) thus

h

f

i

∆(

x

) =

h

f

i

(∆

<

(

x

)

t

{

x

} t

>

(

x

))

<

f

(

x

)

t {

f

(

x

)

} t

>

f

(

x

) = ∆(

f

(

x

)).

Theorem

2177

.

Let function

f

:

A

B

where

A, B

P

R

.

1

.

f

is locally monotone and

f

C(

A, B

) iff

f

C(

A, B

)

C(

ι

A

|

R

|

, ι

B

|

R

|

)

iff

f

C(

A, B

)

C(

ι

A

|

R

|

>

, ι

B

|

R

|

) iff

f

C(

ι

A

|

R

|

, ι

B

|

R

|

)

C(

ι

A

|

R

|

, ι

B

|

R

|

).

2

.

f

is locally strictly monotone and

C(

A, B

) iff

f

C(

A, B

)

C(

ι

A

|

R

|

>

, ι

B

|

R

|

>

) iff

f

C(

ι

A

|

R

|

>

, ι

B

|

R

|

>

)

C(

ι

A

|

R

|

<

, ι

B

|

R

|

<

).

Proof.

By the lemma it is (strictly) monotone on each connected component.

See also related math.SE questions:

1

.

http://math.stackexchange.com/q/1473668/4876

2

.

http://math.stackexchange.com/a/1872906/4876

3

.

http://math.stackexchange.com/q/1875975/4876