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Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions. Another characterization states that the functions of bounded variation on a compact interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone ...
The p variation of a function decreases with p. If f has finite p-variation and g is an α-Hölder continuous function, then has finite -variation. The case when p is one is called total variation, and functions with a finite 1-variation are called bounded variation functions.
[5] [6] [7] Because of the integration by part formula, the integral exists also if the condition on f and g are inversed, that is, if f is of bounded variation and g is continuous. A function g is of bounded variation if and only if it is the difference between two (bounded) monotone functions. If g is not of bounded variation, then there will ...
The set of discontinuities of a regulated function of bounded variation BV is countable for such functions have only jump-type of discontinuities. To see this it is sufficient to note that given ϵ > 0 {\displaystyle \epsilon >0} , the set of points at which the right and left limits differ by more than ϵ {\displaystyle \epsilon } is finite.
In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician Eduard Helly. A more general version of the theorem asserts compactness of the space BV loc of functions locally of bounded total variation that are uniformly bounded at a point.
The bounded interval (a, b) may be replaced with an unbounded interval (-∞, b), (a, ∞) or (-∞, ∞) provided that U and V are of finite variation on this unbounded interval. Complex-valued functions may be used as well. An alternative result, of significant importance in the theory of stochastic calculus is the following.
The basic concept of a Caccioppoli set was first introduced by the Italian mathematician Renato Caccioppoli in the paper (Caccioppoli 1927): considering a plane set or a surface defined on an open set in the plane, he defined their measure or area as the total variation in the sense of Tonelli of their defining functions, i.e. of their parametric equations, provided this quantity was bounded.
The Cantor function is also a standard example of a function with bounded variation but, as mentioned above, is not absolutely continuous. However, every absolutely continuous function is continuous with bounded variation. The Cantor function is non-decreasing, and so in particular its graph defines a rectifiable curve.