<abstract xmlns="http://www.w3.org/1999/xhtml">

It is well known that if the surface <italic>f</italic> : [−1, 1] × [−1, 1] → ℝ has a finite area, then the total variations of both sections <italic>fx</italic>(<italic>y</italic>)= <italic>f</italic> (<italic>x, y</italic>)and <italic>f y</italic>(<italic>x</italic>) = <italic>f</italic> (<italic>x, y</italic>)of <italic>f</italic> 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].
</abstract>

It is well known that if the surface f : [−1, 1] × [−1, 1] → ℝ has a finite area, then the total variations of both sections fx(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].

Remark on a Theorem of Tonelli

Faculty of Mathematics and Computer Science University

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Remark on a Theorem of Tonelli"}

It is well known that if the surface f : [−1, 1] × [−1, 1] → ℝ has a finite area, then the total variations of both sections fx(y)= f (x, y)and f y(x) = f (x,...

Remark on a Theorem of Tonelli

Published Online: Nov 29, 2022

Page range: 89 - 92

Received: Apr 20, 2022

DOI: https://doi.org/10.2478/tmmp-2022-0006

Keywords
surface area, bounded variation, Baire category

© 2022 Władysław Wilczyński, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Remark on a Theorem of Tonelli

Published Online: Nov 29, 2022

Page range: 89 - 92

Received: Apr 20, 2022

DOI: https://doi.org/10.2478/tmmp-2022-0006

Keywordssurface area, bounded variation, Baire category

© 2022 Władysław Wilczyński, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
surface area, bounded variation, Baire category