<s>
In	O
mathematics	O
,	O
the	O
uniform	B-Algorithm
limit	I-Algorithm
theorem	I-Algorithm
states	O
that	O
the	O
uniform	B-Algorithm
limit	I-Algorithm
of	O
any	O
sequence	O
of	O
continuous	O
functions	O
is	O
continuous	O
.	O
</s>
<s>
According	O
to	O
the	O
uniform	B-Algorithm
limit	I-Algorithm
theorem	I-Algorithm
,	O
if	O
each	O
of	O
the	O
functions	O
n	O
is	O
continuous	O
,	O
then	O
the	O
limit	O
must	O
be	O
continuous	O
as	O
well	O
.	O
</s>
<s>
This	O
theorem	O
does	O
not	O
hold	O
if	O
uniform	B-Algorithm
convergence	I-Algorithm
is	O
replaced	O
by	O
pointwise	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
In	O
terms	O
of	O
function	B-Algorithm
spaces	I-Algorithm
,	O
the	O
uniform	B-Algorithm
limit	I-Algorithm
theorem	I-Algorithm
says	O
that	O
the	O
space	O
C(X,Y )	O
of	O
all	O
continuous	O
functions	O
from	O
a	O
topological	O
space	O
X	O
to	O
a	O
metric	O
space	O
Y	O
is	O
a	O
closed	O
subset	O
of	O
YX	O
under	O
the	O
uniform	O
metric	O
.	O
</s>
<s>
The	O
uniform	B-Algorithm
limit	I-Algorithm
theorem	I-Algorithm
also	O
holds	O
if	O
continuity	O
is	O
replaced	O
by	O
uniform	O
continuity	O
.	O
</s>
<s>
Since	O
the	O
sequence	O
of	O
functions	O
(	O
fn	O
)	O
converges	B-Algorithm
uniformly	I-Algorithm
to	O
f	O
by	O
hypothesis	O
,	O
there	O
exists	O
a	O
natural	O
number	O
N	O
such	O
that	O
:	O
</s>
<s>
There	O
are	O
also	O
variants	O
of	O
the	O
uniform	B-Algorithm
limit	I-Algorithm
theorem	I-Algorithm
that	O
are	O
used	O
in	O
complex	O
analysis	O
,	O
albeit	O
with	O
modified	O
assumptions	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
sequence	O
of	O
holomorphic	O
functions	O
that	O
converges	B-Algorithm
uniformly	I-Algorithm
to	O
a	O
function	O
on	O
every	O
compact	O
subset	O
of	O
.	O
</s>
<s>
Then	O
is	O
holomorphic	O
in	O
,	O
and	O
moreover	O
,	O
the	O
sequence	O
of	O
derivatives	O
converges	B-Algorithm
uniformly	I-Algorithm
to	O
on	O
every	O
compact	O
subset	O
of	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
sequence	O
of	O
univalent	O
functions	O
that	O
converges	B-Algorithm
uniformly	I-Algorithm
to	O
a	O
function	O
.	O
</s>
