<s>
In	O
mathematics	O
,	O
Hartogs	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
is	O
a	O
fundamental	O
result	O
of	O
Friedrich	O
Hartogs	O
in	O
the	O
theory	O
of	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
.	O
</s>
<s>
Roughly	O
speaking	O
,	O
it	O
states	O
that	O
a	O
'	O
separately	O
analytic	B-Language
 '	O
function	O
is	O
continuous	O
.	O
</s>
<s>
More	O
precisely	O
,	O
if	O
is	O
a	O
function	O
which	O
is	O
analytic	B-Language
in	O
each	O
variable	O
zi	O
,	O
1	O
i	O
n	O
,	O
while	O
the	O
other	O
variables	O
are	O
held	O
constant	O
,	O
then	O
F	O
is	O
a	O
continuous	O
function	O
.	O
</s>
<s>
A	O
corollary	O
is	O
that	O
the	O
function	O
F	O
is	O
then	O
in	O
fact	O
an	O
analytic	B-Language
function	I-Language
in	O
the	O
n-variable	O
sense	O
(	O
i.e.	O
</s>
<s>
Therefore	O
,	O
'	O
separate	O
analyticity	O
 '	O
and	O
'	O
analyticity	O
 '	O
are	O
coincident	O
notions	O
,	O
in	O
the	O
theory	O
of	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
.	O
</s>
<s>
is	O
differentiable	O
(	O
or	O
even	O
analytic	B-Language
)	O
in	O
each	O
variable	O
separately	O
,	O
it	O
is	O
not	O
true	O
that	O
will	O
necessarily	O
be	O
continuous	O
.	O
</s>
<s>
(	O
Indeed	O
,	O
the	O
limits	B-Algorithm
along	O
the	O
lines	O
and	O
are	O
not	O
equal	O
,	O
so	O
there	O
is	O
no	O
way	O
to	O
extend	O
the	O
definition	O
of	O
to	O
include	O
the	O
origin	O
and	O
have	O
the	O
function	O
be	O
continuous	O
there	O
.	O
)	O
</s>
