total variation
Let be a function mapping an interval to a metric space . We say that is of bounded variation if there is a constant such that, for each partition of ,
The total variation of is defined by
It can be shown that, if is either or , every continuously differentiable (or piecewise continuously differentiable) function is of bounded variation (http://planetmath.org/ContinuousDerivativeImpliesBoundedVariation), and
Also, if is of bounded variation and is continuous, then the Riemann-Stieltjes integral is finite.
If is also continuous, it is said to be a rectifiable path, and is the length of its trace.
If , it can be shown that is of bounded variation if and only if it is the difference of two monotonic functions.
Title | total variation |
Canonical name | TotalVariation |
Date of creation | 2013-03-22 13:26:09 |
Last modified on | 2013-03-22 13:26:09 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 8 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 26A45 |
Classification | msc 26B30 |
Related topic | BVFunction |
Related topic | IntegralRepresentationOfLengthOfSmoothCurve |
Related topic | OscillationOfAFunction |
Defines | bounded variation |
Defines | rectifiable path |