On the points of regularity of multivariate functions of bounded variation

In the one-dimensional case it is well-known that functions of bounded variation on R possess at most a countable number of non-regular points. In this paper we will show that multivariate functions f : Rn → R of bounded variation satisfying the condition lim|x|→∞ f (x) = 0 are non-regular at most on a subset of Rn of Lebesgue measure zero. Moreover, we will point out that this result is best possible.