Bounded Variation

A function f(x) is said to have bounded variation if, over the closed interval x in [a,b], there exists an M such that


for all a<x_1<x_2<...<x_(n-1)<b.

The space of functions of bounded variation is denoted "BV," and has the seminorm


where phi ranges over all compactly supported functions bounded by -1 and 1. The seminorm is equal to the supremum over all sums above, and is also equal to int|df/dx|dx (when this expression makes sense).


On the interval [0,1], the function x^2sin(1/x) (purple) is of bounded variation, but xsin1/x (red) is not. More generally, a function f is locally of bounded variation in a domain U if f is locally integrable, f in L_(loc)^1, and for all open subsets W, with compact closure in U, and all smooth vector fields g compactly supported in W,


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

Such functions form the space BV_(loc)(U). They may not be differentiable, but by the Riesz representation theorem, the derivative of a BV_(loc)-function f is a regular Borel measure Df. Functions of bounded variation also satisfy a compactness theorem.

Given a sequence f_n of functions in BV_(loc)(U), such that


that is the total variation of the functions is bounded, in any compactly supported open subset W, there is a subsequence f_(n_k) which converges to a function f in BV_(loc) in the topology of L_(loc)^1. Moreover, the limit satisfies

 int_W|Df|dx<=lim infint_W|Df_(n_k)|dx.

They also satisfy a version of Poincaré's lemma.

See also

Compact Support, Differentiable, Weakly 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.

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