Bounded variation

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In mathematics, given f, a real-valued function on the interval [a, b] on the real line, the total variation of f on that interval is

$\mathrm{sup}_P \sum_i | f(x_{i+1})-f(x_i) |, \,$

the supremum running over all partitions P = { x₀, ..., x_n } of the interval [a, b]. In effect, the total variation is the vertical component of the arc-length of the graph of f (if f were continuous). The function f is said to be of bounded variation precisely if the total variation of f is finite.

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 closed interval are exactly those f which can be written as a difference g − h, where both g and h are monotone.

1 Applications (in mathematics)
2 Extension
3 Example
4 Reference

[edit] Applications (in mathematics)

Functions of bounded variation have been studied in connection with the set of discontinuities of functions and differentiability of real functions, and the following results are well-known. If $f$ is a real function of bounded variation on an interval [a, b] then

$f$ is continuous except at most on a countable set;
$f$ has one-sided limits everywhere (limits from the left everywhere in (a, b], and from the right everywhere in [a,b) );
the derivative $f'(x)$ exists almost everywhere (i.e. except for a set of measure zero).