Remark on a Theorem of Tonelli

Abstract

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].