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

<math>\mathrm{sup}_P \sum_i | f(x_{i+1})-f(x_i) |, \,</math>

the supremum running over all partitions P = { x₁, ..., 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. 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.

For functions f whose domains are subsets of Rⁿ, f has bounded variation if its weak derivative is a finite measure.

Example

The function

<math>f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x \sin(1/x), & \mbox{if } x \neq 0 \end{cases} </math>

is not of bounded variation on the interval <math> [0, 2/\pi]</math>. In the same time, the function

<math>f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x^2 \sin(1/x), & \mbox{if } x \neq 0 \end{cases} </math>

is of bounded variation on the interval <math> [0,2/\pi]</math>.Template:Mathanalysis-stub

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