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

$\backslash mathrm\{sup\}\_P\; \backslash sum\_i\; |\; f(x\_\{i+1\})-f(x\_i)\; |,\; \backslash ,$

the supremum running over all partitions P = { x₀, ..., x_n } of the interval 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.

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 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 $f\text{'}(x)$ exists almost everywhere (i.e. except for a set of measure zero).

Extension

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

Example

The function

is not of bounded variation on the interval $2/\backslash pi$. In the same time, the function

is of bounded variation on the interval $*$.

Reference

Mathematical analysis

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