<s>
In	O
the	O
mathematical	O
field	O
of	O
analysis	O
,	O
uniform	B-Algorithm
convergence	I-Algorithm
is	O
a	O
mode	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
of	O
functions	O
stronger	O
than	O
pointwise	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
A	O
sequence	O
of	O
functions	O
converges	B-Algorithm
uniformly	I-Algorithm
to	O
a	O
limiting	O
function	O
on	O
a	O
set	O
if	O
,	O
given	O
any	O
arbitrarily	O
small	O
positive	O
number	O
,	O
a	O
number	O
can	O
be	O
found	O
such	O
that	O
each	O
of	O
the	O
functions	O
differs	O
from	O
by	O
no	O
more	O
than	O
at	O
every	O
point	O
in	O
.	O
</s>
<s>
In	O
contrast	O
,	O
pointwise	B-Algorithm
convergence	I-Algorithm
of	O
to	O
merely	O
guarantees	O
that	O
for	O
any	O
given	O
in	O
advance	O
,	O
we	O
can	O
find	O
(	O
can	O
depend	O
on	O
the	O
value	O
of	O
)	O
so	O
that	O
,	O
for	O
that	O
particular	O
,	O
falls	O
within	O
of	O
whenever	O
.	O
</s>
<s>
The	O
difference	O
between	O
uniform	B-Algorithm
convergence	I-Algorithm
and	O
pointwise	B-Algorithm
convergence	I-Algorithm
was	O
not	O
fully	O
appreciated	O
early	O
in	O
the	O
history	O
of	O
calculus	O
,	O
leading	O
to	O
instances	O
of	O
faulty	O
reasoning	O
.	O
</s>
<s>
The	O
concept	O
,	O
which	O
was	O
first	O
formalized	O
by	O
Karl	O
Weierstrass	O
,	O
is	O
important	O
because	O
several	O
properties	O
of	O
the	O
functions	O
,	O
such	O
as	O
continuity	O
,	O
Riemann	O
integrability	O
,	O
and	O
,	O
with	O
additional	O
hypotheses	O
,	O
differentiability	O
,	O
are	O
transferred	O
to	O
the	O
limit	B-Algorithm
if	O
the	O
convergence	B-Algorithm
is	O
uniform	O
,	O
but	O
not	O
necessarily	O
if	O
the	O
convergence	B-Algorithm
is	O
not	O
uniform	O
.	O
</s>
<s>
Completely	O
standard	O
notions	O
of	O
convergence	B-Algorithm
did	O
not	O
exist	O
at	O
the	O
time	O
,	O
and	O
Cauchy	O
handled	O
convergence	B-Algorithm
using	O
infinitesimal	O
methods	O
.	O
</s>
<s>
When	O
put	O
into	O
the	O
modern	O
language	O
,	O
what	O
Cauchy	O
proved	O
is	O
that	O
a	O
uniformly	B-Algorithm
convergent	I-Algorithm
sequence	O
of	O
continuous	O
functions	O
has	O
a	O
continuous	O
limit	B-Algorithm
.	O
</s>
<s>
The	O
failure	O
of	O
a	O
merely	O
pointwise-convergent	O
limit	B-Algorithm
of	O
continuous	O
functions	O
to	O
converge	O
to	O
a	O
continuous	O
function	O
illustrates	O
the	O
importance	O
of	O
distinguishing	O
between	O
different	O
types	O
of	O
convergence	B-Algorithm
when	O
handling	O
sequences	O
of	O
functions	O
.	O
</s>
<s>
The	O
term	O
uniform	B-Algorithm
convergence	I-Algorithm
was	O
probably	O
first	O
used	O
by	O
Christoph	O
Gudermann	O
,	O
in	O
an	O
1838	O
paper	O
on	O
elliptic	O
functions	O
,	O
where	O
he	O
employed	O
the	O
phrase	O
"	O
convergence	B-Algorithm
in	O
a	O
uniform	O
way	O
"	O
when	O
the	O
"	O
mode	B-Algorithm
of	I-Algorithm
convergence	I-Algorithm
"	O
of	O
a	O
series	O
is	O
independent	O
of	O
the	O
variables	O
and	O
While	O
he	O
thought	O
it	O
a	O
"	O
remarkable	O
fact	O
"	O
when	O
a	O
series	O
converged	O
in	O
this	O
way	O
,	O
he	O
did	O
not	O
give	O
a	O
formal	O
definition	O
,	O
nor	O
use	O
the	O
property	O
in	O
any	O
of	O
his	O
proofs	O
.	O
</s>
<s>
G	O
.	O
H	O
.	O
Hardy	O
compares	O
the	O
three	O
definitions	O
in	O
his	O
paper	O
"	O
Sir	O
George	O
Stokes	O
and	O
the	O
concept	O
of	O
uniform	B-Algorithm
convergence	I-Algorithm
"	O
and	O
remarks	O
:	O
"	O
Weierstrass	O
's	O
discovery	O
was	O
the	O
earliest	O
,	O
and	O
he	O
alone	O
fully	O
realized	O
its	O
far-reaching	O
importance	O
as	O
one	O
of	O
the	O
fundamental	O
ideas	O
of	O
analysis.	O
"	O
</s>
<s>
We	O
first	O
define	O
uniform	B-Algorithm
convergence	I-Algorithm
for	O
real-valued	O
functions	O
,	O
although	O
the	O
concept	O
is	O
readily	O
generalized	O
to	O
functions	O
mapping	O
to	O
metric	O
spaces	O
and	O
,	O
more	O
generally	O
,	O
uniform	O
spaces	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
The	O
notation	O
for	O
uniform	B-Algorithm
convergence	I-Algorithm
of	O
to	O
is	O
not	O
quite	O
standardized	O
and	O
different	O
authors	O
have	O
used	O
a	O
variety	O
of	O
symbols	O
,	O
including	O
(	O
in	O
roughly	O
decreasing	O
order	O
of	O
popularity	O
)	O
:	O
</s>
<s>
to	O
indicate	O
that	O
convergence	B-Algorithm
is	O
uniform	O
.	O
</s>
<s>
(	O
In	O
contrast	O
,	O
the	O
expression	O
on	O
without	O
an	O
adverb	O
is	O
taken	O
to	O
mean	O
pointwise	B-Algorithm
convergence	I-Algorithm
on	O
:	O
for	O
all	O
,	O
as	O
.	O
)	O
</s>
<s>
The	O
sequence	O
is	O
said	O
to	O
be	O
locally	B-Algorithm
uniformly	I-Algorithm
convergent	I-Algorithm
with	O
limit	B-Algorithm
if	O
is	O
a	O
metric	O
space	O
and	O
for	O
every	O
,	O
there	O
exists	O
an	O
such	O
that	O
converges	B-Algorithm
uniformly	I-Algorithm
on	O
It	O
is	O
clear	O
that	O
uniform	B-Algorithm
convergence	I-Algorithm
implies	O
local	B-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
,	O
which	O
implies	O
pointwise	B-Algorithm
convergence	I-Algorithm
.	O
</s>
