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
 |
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
 |
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=28014
Total variation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Total_variation_of_a_function&oldid=28014
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article