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16.4. GENERALIZED LIMIT

229

Reversely

h

µ

i

{

x

} ×

FCD

h

ν

i

{

y

}

= (

h

µ

i

{

x

} ×

FCD

h

ν

i

{

y

}

)

◦ ↑

e

where

e

is the

identify element of

G

.

Proposition

1191

.

τ

(

y

) = xlim(

h

µ

i

{

x

FCD

Base(Ob

ν

)

{

y

}

) (for every

x

).

Informally: Every

τ

(

y

) is a generalized limit of a constant funcoid.

Proof.

xlim(

h

µ

i

{

x

FCD

Base(Ob

ν

)

{

y

}

) =

ν

(

h

µ

i

{

x

FCD

Base(Ob

ν

)

{

y

}

)

◦ ↑

r

r

G

=

(

h

µ

i

{

x

} ×

FCD

h

ν

i

{

y

}

)

◦ ↑

r

r

G

=

τ

(

y

)

.

Theorem

1192

.

If

f

|

h

µ

i

{

x

}

C(

µ

;

ν

) and

h

µ

i

{

x

} w↑

Ob

µ

{

x

}

then xlim

x

f

=

τ

(

f x

).

Proof.

f

|

h

µ

i

{

x

}

µ

v

ν

f

|

h

µ

i

{

x

}

v

ν

f

; thus

h

f

ih

µ

i

{

x

} v h

ν

ih

f

i

{

x

}

;

consequently we have

ν

w h

ν

ih

f

i

{

x

} ×

FCD

h

ν

ih

f

i

{

x

} w h

f

ih

µ

i

{

x

} ×

FCD

h

ν

ih

f

i

{

x

}

.

ν

f

|

h

µ

i

{

x

}

w

(

h

f

ih

µ

i

{

x

} ×

FCD

h

ν

ih

f

i

{

x

}

)

f

|

h

µ

i

{

x

}

=

(

f

|

h

µ

i

{

x

}

)

1

h

f

ih

µ

i

{

x

} ×

FCD

h

ν

ih

f

i

{

x

} w

D

id

FCD

dom

f

|

h

µ

i∗ {

x

}

E

h

µ

i

{

x

} ×

FCD

h

ν

ih

f

i

{

x

} w

dom

f

|

h

µ

i

{

x

}

×

FCD

h

ν

ih

f

i

{

x

}

=

h

µ

i

{

x

} ×

FCD

h

ν

ih

f

i

{

x

}

.

im(

ν

f

|

h

µ

i

{

x

}

) =

h

ν

ih

f

i

{

x

}

;

ν

f

|

h

µ

i

{

x

}

v

h

µ

i

{

x

} ×

FCD

im(

ν

f

|

h

µ

i

{

x

}

) =

h

µ

i

{

x

} ×

FCD

h

ν

ih

f

i

{

x

}

.

So

ν

f

|

h

µ

i

{

x

}

=

h

µ

i

{

x

} ×

FCD

h

ν

ih

f

i

{

x

}

.

Thus xlim

x

f

=

n

(

h

µ

i

{

x

FCD

h

ν

ih

f

i

{

x

}

)

◦↑

r

r

G

o

=

τ

(

f x

).

Remark

1193

.

Without the requirement of

h

µ

i

{

x

} w↑

Ob

µ

{

x

}

the last theo-

rem would not work in the case of removable singularity.

Theorem

1194

.

Let

ν

v

ν

ν

. If

f

|

h

µ

i

{

x

}

ν

→↑

Ob

µ

{

y

}

then xlim

x

f

=

τ

(

y

).