A function is said to have bounded variation if, over the closed
interval ,
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 ,
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 smoothvector
fieldscompactly supported in ,

(3)

div denotes divergence and is a constant which only depends on the choice of and .

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