<s>
In	O
mathematics	O
,	O
the	O
oscillation	B-Algorithm
of	O
a	O
function	O
or	O
a	O
sequence	O
is	O
a	O
number	O
that	O
quantifies	O
how	O
much	O
that	O
sequence	O
or	O
function	O
varies	O
between	O
its	O
extreme	O
values	O
as	O
it	O
approaches	O
infinity	O
or	O
a	O
point	O
.	O
</s>
<s>
As	O
is	O
the	O
case	O
with	O
limits	O
,	O
there	O
are	O
several	O
definitions	O
that	O
put	O
the	O
intuitive	O
concept	O
into	O
a	O
form	O
suitable	O
for	O
a	O
mathematical	O
treatment	O
:	O
oscillation	B-Algorithm
of	O
a	O
sequence	O
of	O
real	O
numbers	O
,	O
oscillation	B-Algorithm
of	O
a	O
real-valued	O
function	O
at	O
a	O
point	O
,	O
and	O
oscillation	B-Algorithm
of	O
a	O
function	O
on	O
an	O
interval	O
(	O
or	O
open	O
set	O
)	O
.	O
</s>
<s>
The	O
oscillation	B-Algorithm
of	O
that	O
sequence	O
is	O
defined	O
as	O
the	O
difference	O
(	O
possibly	O
infinite	O
)	O
between	O
the	O
limit	B-Algorithm
superior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
inferior	I-Algorithm
of	O
:	O
</s>
<s>
The	O
oscillation	B-Algorithm
is	O
zero	O
if	O
and	O
only	O
if	O
the	O
sequence	O
converges	B-Algorithm
.	O
</s>
<s>
The	O
oscillation	B-Algorithm
of	O
on	O
an	O
interval	O
in	O
its	O
domain	O
is	O
the	O
difference	O
between	O
the	O
supremum	O
and	O
infimum	O
of	O
:	O
</s>
<s>
The	O
oscillation	B-Algorithm
of	O
a	O
function	O
of	O
a	O
real	O
variable	O
at	O
a	O
point	O
is	O
defined	O
as	O
the	O
limit	B-Algorithm
as	O
of	O
the	O
oscillation	B-Algorithm
of	O
on	O
an	O
-neighborhood	O
of	O
:	O
</s>
<s>
This	O
is	O
the	O
same	O
as	O
the	O
difference	O
between	O
the	O
limit	B-Algorithm
superior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
inferior	I-Algorithm
of	O
the	O
function	O
at	O
,	O
provided	O
the	O
point	O
is	O
not	O
excluded	O
from	O
the	O
limits	O
.	O
</s>
<s>
has	O
oscillation	B-Algorithm
∞	O
at	O
=	O
0	O
,	O
and	O
oscillation	B-Algorithm
0	O
at	O
other	O
finite	O
and	O
at	O
−∞	O
and	O
+∞	O
.	O
</s>
<s>
(	O
the	O
topologist	O
's	O
sine	O
curve	O
)	O
has	O
oscillation	B-Algorithm
2	O
at	O
=	O
0	O
,	O
and	O
0	O
elsewhere	O
.	O
</s>
<s>
has	O
oscillation	B-Algorithm
0	O
at	O
every	O
finite	O
,	O
and	O
2	O
at	O
−∞	O
and	O
+∞	O
.	O
</s>
<s>
or	O
1	O
,	O
-1	O
,	O
1	O
,	O
-1	O
,	O
1	O
,	O
1	O
...	O
has	O
oscillation	B-Algorithm
2	O
.	O
</s>
<s>
In	O
the	O
last	O
example	O
the	O
sequence	O
is	O
periodic	O
,	O
and	O
any	O
sequence	O
that	O
is	O
periodic	O
without	O
being	O
constant	O
will	O
have	O
non-zero	O
oscillation	B-Algorithm
.	O
</s>
<s>
However	O
,	O
non-zero	O
oscillation	B-Algorithm
does	O
not	O
usually	O
indicate	O
periodicity	O
.	O
</s>
<s>
Oscillation	B-Algorithm
can	O
be	O
used	O
to	O
define	O
continuity	O
of	O
a	O
function	O
,	O
and	O
is	O
easily	O
equivalent	O
to	O
the	O
usual	O
ε-δ	O
definition	O
(	O
in	O
the	O
case	O
of	O
functions	O
defined	O
everywhere	O
on	O
the	O
real	O
line	O
)	O
:	O
a	O
function	O
ƒ	O
is	O
continuous	O
at	O
a	O
point	O
x0	O
if	O
and	O
only	O
if	O
the	O
oscillation	B-Algorithm
is	O
zero	O
;	O
in	O
symbols	O
,	O
A	O
benefit	O
of	O
this	O
definition	O
is	O
that	O
it	O
quantifies	O
discontinuity	O
:	O
the	O
oscillation	B-Algorithm
gives	O
how	O
much	O
the	O
function	O
is	O
discontinuous	O
at	O
a	O
point	O
.	O
</s>
<s>
in	O
a	O
removable	O
discontinuity	O
,	O
the	O
distance	O
that	O
the	O
value	O
of	O
the	O
function	O
is	O
off	O
by	O
is	O
the	O
oscillation	B-Algorithm
;	O
</s>
<s>
in	O
a	O
jump	O
discontinuity	O
,	O
the	O
size	O
of	O
the	O
jump	O
is	O
the	O
oscillation	B-Algorithm
(	O
assuming	O
that	O
the	O
value	O
at	O
the	O
point	O
lies	O
between	O
these	O
limits	O
from	O
the	O
two	O
sides	O
)	O
;	O
</s>
<s>
in	O
an	O
essential	O
discontinuity	O
,	O
oscillation	B-Algorithm
measures	O
the	O
failure	O
of	O
a	O
limit	B-Algorithm
to	O
exist	O
.	O
</s>
<s>
This	O
definition	O
is	O
useful	O
in	O
descriptive	O
set	O
theory	O
to	O
study	O
the	O
set	O
of	O
discontinuities	O
and	O
continuous	O
points	O
–	O
the	O
continuous	O
points	O
are	O
the	O
intersection	O
of	O
the	O
sets	O
where	O
the	O
oscillation	B-Algorithm
is	O
less	O
than	O
ε	O
(	O
hence	O
a	O
Gδ	O
set	O
)	O
–	O
and	O
gives	O
a	O
very	O
quick	O
proof	O
of	O
one	O
direction	O
of	O
the	O
Lebesgue	O
integrability	O
condition	O
.	O
</s>
<s>
The	O
oscillation	B-Algorithm
is	O
equivalent	O
to	O
the	O
ε-δ	O
definition	O
by	O
a	O
simple	O
re-arrangement	O
,	O
and	O
by	O
using	O
a	O
limit	B-Algorithm
(	O
lim	B-Algorithm
sup	I-Algorithm
,	O
lim	B-Algorithm
inf	I-Algorithm
)	O
to	O
define	O
oscillation	B-Algorithm
:	O
if	O
(	O
at	O
a	O
given	O
point	O
)	O
for	O
a	O
given	O
ε0	O
there	O
is	O
no	O
δ	O
that	O
satisfies	O
the	O
ε-δ	O
definition	O
,	O
then	O
the	O
oscillation	B-Algorithm
is	O
at	O
least	O
ε0	O
,	O
and	O
conversely	O
if	O
for	O
every	O
ε	O
there	O
is	O
a	O
desired	O
δ	O
,	O
the	O
oscillation	B-Algorithm
is	O
0	O
.	O
</s>
<s>
The	O
oscillation	B-Algorithm
definition	O
can	O
be	O
naturally	O
generalized	O
to	O
maps	O
from	O
a	O
topological	O
space	O
to	O
a	O
metric	O
space	O
.	O
</s>
