A function 
is said to have bounded variation if, over the closed
interval ![x in [a,b]](/images/equations/BoundedVariation/Inline2.svg)
,
there exists an 
such that
 |
(1)
|
for all 
.
The space of functions of bounded variation is denoted "BV," and has the seminorm
 |
(2)
|
where 
ranges over all compactly supported functions
bounded by 
and 1. The seminorm is equal to the supremum over all
sums above, and is also equal to 
(when this expression makes sense).
On the interval ![[0,1]](/images/equations/BoundedVariation/Inline8.svg)
,
the function 
(purple) is of bounded variation, but 
(red) is not. More generally, a function 
is locally of bounded variation in a domain 
if 
is locally integrable,
,
and for all open subsets 
, with compact closure in
,
and all smooth vector
fields 
compactly supported in 
,
 |
(3)
|
div denotes divergence and 
is a constant which only depends on the choice of 
and 
.
Such functions form the space 
. They may not be differentiable,
but by the Riesz representation theorem,
the derivative of a 
-function
is a regular Borel measure 
. Functions of bounded variation also satisfy a compactness
theorem.
Given a sequence 
of functions in 
,
such that
![sup_(n)[||f_n||_(L^1(W))+int_W|Df_n|dx]<infty,](/images/equations/BoundedVariation/NumberedEquation4.svg) |
(4)
|
that is the total variation of the functions is bounded, in any compactly supported open subset
,
there is a subsequence 
which converges to a function 
in the topology of 
. Moreover, the limit satisfies
 |
(5)
|
They also satisfy a version of Poincaré's lemma.
