Difference between revisions of "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 $$N(y,f) = +\infty,$$
$$N(y,f) = 0.$$
The function $N(y,f)$ was defined by S. Banach  (see also ). He proved that the indicatrix $N(y,f)$ of any continuous function $f(x)$ in the interval $[a,b]$ is a function of 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 . The concept of a Banach indicatrix was employed to define the variation of functions in several variables , .