# Banach indicatrix

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multiplicity function, of a continuous function , An integer-valued function , , equal to the number of roots of the equation . If, for a given value of , this equation has an infinite number of roots, then and if it has no roots, then The function was defined by S. Banach  (see also ). He proved that the indicatrix of any continuous function in the interval is a function of Baire class not higher than 2, and (*)

where is the variation of on . Thus, equation (*) can be considered as the definition of the variation of a continuous function . The Banach indicatrix is also defined (preserving equation (*)) for functions with discontinuities of the first kind . The concept of a Banach indicatrix was employed to define the variation of functions in several variables , .

How to Cite This Entry:
Banach indicatrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach_indicatrix&oldid=14656
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article