Remark on a theorem of Tonelli

Year, pages: 2022, 89 - 92

It is well known that if the surface $f:[-1,1]×[-1,1]\to\mathbb{R}$ has a finite area, then the total variations of both sections $f_x(y)=f(x,y)$ and $f^y(x)=f(x,y)$ of $f$ are finite almost everywhere in $[-1,1]$. In the note it is proved that these variations can be infinite on residual subsets of $[-1,1]$.

Wilczyński, W. 2022. Remark on a theorem of Tonelli. In Tatra Mountains Mathematical Publications, vol. 81, no.1, pp. 89-92. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2022-0006

Wilczyński, W. (2022). Remark on a theorem of Tonelli. Tatra Mountains Mathematical Publications, 81(1), 89-92. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2022-0006

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava

https://creativecommons.org/licenses/by-nc-nd/4.0/