Continuity-Bounded Variation

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 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

(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.