Remark on a theorem of Tonelli

Abstract

It is well known that if the surface
$f:[-1,1]\times[-1,1]\to\mathbb{R}$ 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 $[-1,1]$.