Total variation of a function
From Encyclopedia of Mathematics
The same as the variation of a function of one variable. The total variation of a real-valued function is the sum of its positive variation (cf. Positive variation of a function) and negative variation (cf. Negative variation of a function).
Comments
If is a complex-valued function on
, then its total variation over
is the number
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If is also continuous, then this number is the same as the length of the arc in the complex plane that is parametrized by
. If
is absolutely continuous on
, then
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References
[a1] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) |
How to Cite This Entry:
Total variation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Total_variation_of_a_function&oldid=17306
Total variation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Total_variation_of_a_function&oldid=17306
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article