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# 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.

Compact Support, Differentiable

Portions of this entry contributed by Todd Rowland

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## Cite this as:

Rowland, Todd and Weisstein, Eric W. "Bounded Variation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BoundedVariation.html