View source for Banach indicatrix

''multiplicity function, of a continuous function $y=f(x)$, $a\leq x\leq b$''

An integer-valued function $N(y,f)$, $-\infty < y < \infty$, equal to the number of roots of the equation $f(x)=y$. 
If, for a given value of $y$, this equation has an infinite number of roots, then
$$
N(y,f) = +\infty,
$$

and if it has no roots, then

$$
N(y,f) = 0.
$$

The function $N(y,f)$ was defined by S. Banach [[#References|[1]]] (see also [[#References|[2]]]). 
He proved that the indicatrix $N(y,f)$ of any continuous function $f(x)$ in the interval $[a,b]$ is a function of [[Baire classes|Baire class]] not higher than 2, and
\begin{equation}\label{eq1}
V_a^b(f) = \int\limits_{-\infty}^{+\infty} N(y, f) \, dy,
\end{equation}

where $V_a^b(f)$ is the variation of $f(x)$ on $[a,b]$. Thus, equation \eqref{eq1} can be considered as the definition of the variation of a continuous function $f(x)$. The Banach indicatrix is also defined (preserving equation \eqref{eq1}) for functions with discontinuities of the first kind [[#References|[3]]]. The concept of a Banach indicatrix was employed to define the variation of functions in several variables [[#References|[4]]], [[#References|[5]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Banach,   "Sur les lignes rectifiables et les surfaces dont l'aire est finie"  ''Fund. Math.'' , '''7'''  (1925)  pp. 225–236</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,   "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Lozinskii,   "On the Banach indicatrix"  ''Vestnik Leningrad. Univ. Math. Mekh. Astr.'' , '''7''' :  2  pp. 70–87  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.S. Kronrod,   "On functions of two variables"  ''Uspekhi Mat. Nauk'' , '''5''' :  1  (1950)  pp. 24–134  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.G. Vitushkin,   "On higher-dimensional variations" , Moscow  (1955)  (In Russian)</TD></TR></table>



====Comments====
More generally, for any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515018.png" /> define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515019.png" /> analogously. Then, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515020.png" /> be a separable metric space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515021.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515022.png" />-measurable for all Borel subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515024.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515026.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515027.png" /> be the measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515028.png" /> defined by the Carathéodory construction from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515029.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515030.png" /></td> </tr></table>

for every Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515031.png" />. Cf. [[#References|[a1]]], p. 176 ff. For significant extension of (*), cf. [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Federer,   "Geometric measure theory" , Springer  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Federer,   "An analytic characterization of distributions whose partial derivatives are representable by measures"  ''Bull. Amer. Math. Soc.'' , '''60'''  (1954)  pp. 339</TD></TR></table>