On a generalisation of the Banach indicatrix theorem

Abstract

We prove that for any regulated function $f:\left[a,b\right]\rightarrow\mathbb{R}$ and $c\geq 0$ the infimum of the total variations of functions approximating $f$ with accuracy $c/2$ is equal $\int_{\mathbb{R}} n_{c}^{y} \mathrm{d} y,$ where $n_{c}^{y}$ is the number of times that $f$ crosses the interval $[y,y+c].$