On the points of regularity of multivariate functions of bounded variation.

Abstract

In the one-dimensional case it is well-known that functions of bounded variation on $\mathbb{R}$ possess at most a countable number of non-regular points. In this paper we will show that multivariate functions $f:\mathbb{R}^n \rightarrow \mathbb{R} $ of bounded variation satisfying the condition lim$_{|x| \to \infty} f(X) $ are non-regular at most on a subset of $\mathbb{R}^n$ of Lebesgue measure zero. Moreover, we will point out that this result is best possible.