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16.4. GENERALIZED LIMIT

227

16.2. Relationships between convergence and continuity

Lemma

1174

.

Let

µ

,

ν

be endofuncoids,

f

FCD

(Ob

µ

; Ob

ν

),

A ∈

F

(Ob

µ

),

Src

f

= Ob

µ

, Dst

f

= Ob

ν

. If

f

C(

µ

|

A

;

ν

) then

f

|

h

µ

iA

µ

→ h

f

iA ⇔ h

f

µ

|

A

iA v h

ν

f

iA

.

Proof.

f

|

h

µ

iA

µ

→ h

f

iA ⇔

im

f

|

h

µ

iA

v h

ν

ih

f

iA ⇔

h

f

ih

µ

iA v h

ν

ih

f

iA ⇔ h

f

µ

iA v h

ν

f

iA ⇔ h

f

µ

|

A

iA v h

ν

f

iA

.

Theorem

1175

.

Let

µ

,

ν

be endofuncoids,

f

FCD

(Ob

µ

; Ob

ν

),

A ∈

F

(Ob

µ

),

Src

f

= Ob

µ

, Dst

f

= Ob

ν

. If

f

C(

µ

|

A

;

ν

) then

f

|

h

µ

iA

ν

→ h

f

iA

.

Proof.

f

|

h

µ

iA

ν

→ h

f

iA ⇔

(by the lemma)

⇔ h

f

µ

|

A

iA v h

ν

f

iA ⇐

f

µ

|

A

v

ν

f

f

C(

µ

|

A

;

ν

)

.

Corollary

1176

.

Let

µ

,

ν

be endofuncoids,

f

FCD

(Ob

µ

; Ob

ν

),

A ∈

F

(Ob

µ

), Src

f

= Ob

µ

, Dst

f

= Ob

ν

. If

f

C(

µ

;

ν

) then

f

|

h

µ

iA

ν

→ h

f

iA

.

Theorem

1177

.

Let

µ

,

ν

be endofuncoids,

f

FCD

(Ob

µ

; Ob

ν

),

A ∈

F

(Ob

µ

)

be an ultrafilter, Src

f

= Ob

µ

, Dst

f

= Ob

ν

.

f

C(

µ

|

A

;

ν

) iff

f

|

h

µ

iA

ν

→ h

f

iA

.

Proof.

f

|

h

µ

iA

ν

→ h

f

iA ⇔

(by the lemma)

⇔ h

f

µ

|

A

iA v h

ν

f

iA ⇔

(used the fact that

A

is an ultrafilter)

f

µ

|

A

v

ν

f

|

A

f

µ

|

A

v

ν

f

f

C(

µ

|

A

;

ν

)

.

16.3. Limit

Definition

1178

.

lim

µ

f

=

a

iff

f

µ

→↑

Src

µ

{

a

}

for a

T

2

-separable funcoid

µ

and a non-empty funcoid

f

such that Dst

f

= Dst

µ

.

It is defined correctly, that is

f

has no more than one limit.

Proof.

Let lim

µ

f

=

a

and lim

µ

f

=

b

. Then im

f

v h

µ

i

{

a

}

and im

f

v

h

µ

i

{

b

}

.

Because

f

6

=

FCD

(Src

f

;Dst

f

)

we have im

f

6

=

F

(Dst

f

)

;

h

µ

i

{

a

} u h

µ

i

{

b

} 6

=

F

(Dst

f

)

;

Src

µ

{

b

} u

µ

1

h

µ

i

{

a

} 6

=

F

(Src

µ

)

;

Src

µ

{

b

} u

µ

1

µ

Src

µ

{

a

} 6

=

F

(Src

µ

)

;

{

a

}

µ

1

µ

{

b

}

. Because

µ

is

T

2

-separable we have

a

=

b

.

Definition

1179

.

lim

µ

B

f

= lim

µ

(

f

|

B

).

Remark

1180

.

We can also in an obvious way define limit of a reloid.

16.4. Generalized limit

FiXme

: Refer readers to http://portonmath.tiddlyspace.com/