On a generalisation of the Banach indicatrix theorem

Abstract

We prove that for any regulated function $f:\left[a,b\right]\to \mathbb{R}$ and $c\ge 0$ 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 $[y,y+c].$

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