Bounded variation
|
|
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, ..., xn } 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. There is another characterisation available in distribution theory; they are the functions whose derivative in the distributional sense is a measure.
Another characterization states that the functions of bounded variation are exactly those f which can be written as a difference g − h, where both g and h are monotone.
Examples
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:Math-stub