<s>
In	O
mathematics	O
,	O
multivalued	B-Algorithm
function	I-Algorithm
,	O
also	O
called	O
multifunction	O
and	O
many-valued	B-Algorithm
function	I-Algorithm
,	O
is	O
a	O
set-valued	O
function	O
with	O
continuity	O
properties	O
that	O
allow	O
considering	O
it	O
locally	O
as	O
an	O
ordinary	O
function	O
.	O
</s>
<s>
Multivalued	B-Algorithm
functions	I-Algorithm
arise	O
commonly	O
in	O
applications	O
of	O
implicit	O
function	O
theorem	O
,	O
since	O
this	O
theorem	O
can	O
be	O
viewed	O
as	O
asserting	O
the	O
existence	O
of	O
a	O
multivalued	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
inverse	O
function	O
of	O
a	O
differentiable	O
function	O
is	O
a	O
multivalued	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
complex	O
logarithm	O
is	O
a	O
multivalued	B-Algorithm
function	I-Algorithm
,	O
as	O
the	O
inverse	O
of	O
the	O
exponential	O
function	O
.	O
</s>
<s>
Another	O
common	O
way	O
for	O
defining	O
a	O
multivalued	B-Algorithm
function	I-Algorithm
is	O
analytic	B-Algorithm
continuation	I-Algorithm
,	O
which	O
generates	O
commonly	O
some	O
monodromy	O
:	O
analytic	B-Algorithm
continuation	I-Algorithm
along	O
a	O
closed	O
curve	O
may	O
generate	O
a	O
final	O
value	O
that	O
differs	O
from	O
the	O
starting	O
value	O
.	O
</s>
<s>
Multivalued	B-Algorithm
functions	I-Algorithm
arise	O
also	O
as	O
solutions	O
of	O
differential	O
equations	O
,	O
where	O
the	O
different	O
values	O
are	O
parametrized	O
by	O
initial	B-Algorithm
conditions	I-Algorithm
.	O
</s>
<s>
The	O
term	O
multivalued	B-Algorithm
function	I-Algorithm
originated	O
in	O
complex	O
analysis	O
,	O
from	O
analytic	B-Algorithm
continuation	I-Algorithm
.	O
</s>
<s>
It	O
often	O
occurs	O
that	O
one	O
knows	O
the	O
value	O
of	O
a	O
complex	O
analytic	B-Language
function	I-Language
in	O
some	O
neighbourhood	O
of	O
a	O
point	O
.	O
</s>
<s>
In	O
such	O
a	O
situation	O
,	O
one	O
may	O
extend	O
the	O
domain	O
of	O
the	O
single-valued	B-Algorithm
function	I-Algorithm
along	O
curves	O
in	O
the	O
complex	O
plane	O
starting	O
at	O
.	O
</s>
<s>
In	O
doing	O
so	O
,	O
one	O
finds	O
that	O
the	O
value	O
of	O
the	O
extended	O
function	O
at	O
a	O
point	O
depends	O
on	O
the	O
chosen	O
curve	O
from	O
to	O
;	O
since	O
none	O
of	O
the	O
new	O
values	O
is	O
more	O
natural	O
than	O
the	O
others	O
,	O
all	O
of	O
them	O
are	O
incorporated	O
into	O
a	O
multivalued	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
To	O
define	O
a	O
single-valued	B-Algorithm
function	I-Algorithm
from	O
a	O
complex	O
multivalued	B-Algorithm
function	I-Algorithm
,	O
one	O
may	O
distinguish	O
one	O
of	O
the	O
multiple	O
values	O
as	O
the	O
principal	O
value	O
,	O
producing	O
a	O
single-valued	B-Algorithm
function	I-Algorithm
on	O
the	O
whole	O
plane	O
which	O
is	O
discontinuous	O
along	O
certain	O
boundary	O
curves	O
.	O
</s>
<s>
Alternatively	O
,	O
dealing	O
with	O
the	O
multivalued	B-Algorithm
function	I-Algorithm
allows	O
having	O
something	O
that	O
is	O
everywhere	O
continuous	O
,	O
at	O
the	O
cost	O
of	O
possible	O
value	O
changes	O
when	O
one	O
follows	O
a	O
closed	O
path	O
(	O
monodromy	O
)	O
.	O
</s>
<s>
These	O
problems	O
are	O
resolved	O
in	O
the	O
theory	O
of	O
Riemann	O
surfaces	O
:	O
to	O
consider	O
a	O
multivalued	B-Algorithm
function	I-Algorithm
as	O
an	O
ordinary	O
function	O
without	O
discarding	O
any	O
values	O
,	O
one	O
multiplies	O
the	O
domain	O
into	O
a	O
many-layered	O
covering	O
space	O
,	O
a	O
manifold	B-Architecture
which	O
is	O
the	O
Riemann	O
surface	O
associated	O
to	O
.	O
</s>
<s>
Every	O
real	O
number	O
greater	O
than	O
zero	O
has	O
two	O
real	O
square	O
roots	O
,	O
so	O
that	O
square	O
root	O
may	O
be	O
considered	O
a	O
multivalued	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
complex	O
logarithm	O
function	O
is	O
multiple-valued	B-Algorithm
.	O
</s>
<s>
Inverse	O
trigonometric	O
functions	O
are	O
multiple-valued	B-Algorithm
because	O
trigonometric	O
functions	O
are	O
periodic	O
.	O
</s>
<s>
We	O
can	O
treat	O
arctan	O
as	O
a	O
single-valued	B-Algorithm
function	I-Algorithm
by	O
restricting	O
the	O
domain	O
of	O
tan	O
x	O
to	O
–	O
a	O
domain	O
over	O
which	O
tan	O
x	O
is	O
monotonically	O
increasing	O
.	O
</s>
<s>
The	O
antiderivative	B-Algorithm
can	O
be	O
considered	O
as	O
a	O
multivalued	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
antiderivative	B-Algorithm
of	O
a	O
function	O
is	O
the	O
set	O
of	O
functions	O
whose	O
derivative	O
is	O
that	O
function	O
.	O
</s>
<s>
Inverse	O
hyperbolic	O
functions	O
over	O
the	O
complex	O
domain	O
are	O
multiple-valued	B-Algorithm
because	O
hyperbolic	O
functions	O
are	O
periodic	O
along	O
the	O
imaginary	O
axis	O
.	O
</s>
<s>
Over	O
the	O
reals	O
,	O
they	O
are	O
single-valued	B-Algorithm
,	O
except	O
for	O
arcosh	O
and	O
arsech	O
.	O
</s>
<s>
These	O
are	O
all	O
examples	O
of	O
multivalued	B-Algorithm
functions	I-Algorithm
that	O
come	O
about	O
from	O
non-injective	O
functions	O
.	O
</s>
<s>
Often	O
,	O
the	O
restriction	O
of	O
a	O
multivalued	B-Algorithm
function	I-Algorithm
is	O
a	O
partial	O
inverse	O
of	O
the	O
original	O
function	O
.	O
</s>
<s>
Multivalued	B-Algorithm
functions	I-Algorithm
of	O
a	O
complex	O
variable	O
have	O
branch	O
points	O
.	O
</s>
<s>
Using	O
the	O
branch	O
points	O
,	O
these	O
functions	O
may	O
be	O
redefined	O
to	O
be	O
single-valued	B-Algorithm
functions	I-Algorithm
,	O
by	O
restricting	O
the	O
range	O
.	O
</s>
<s>
In	O
physics	O
,	O
multivalued	B-Algorithm
functions	I-Algorithm
play	O
an	O
increasingly	O
important	O
role	O
.	O
</s>
<s>
They	O
form	O
the	O
mathematical	O
basis	O
for	O
Dirac	O
's	O
magnetic	O
monopoles	O
,	O
for	O
the	O
theory	O
of	O
defects	O
in	O
crystals	O
and	O
the	O
resulting	O
plasticity	O
of	O
materials	O
,	O
for	O
vortices	B-Application
in	O
superfluids	O
and	O
superconductors	O
,	O
and	O
for	O
phase	O
transitions	O
in	O
these	O
systems	O
,	O
for	O
instance	O
melting	O
and	O
quark	O
confinement	O
.	O
</s>
