Distributionally Regulated Functions in Several Variables

Some properties of regulated distributions in several variables are studied.It is shown that S, the set of singular points of a regulated distribution, has measure zero.It is also shown that if S, the closure of S, is at most countable, then the function satisfies the intermediate value property.Furthermore, it is proved that functions g for which f = ∇ 1 • • • ∇ d g is regulated and for which S is at most countable satisfy the mean value property of [R.Estrada, A mean value theorem in several variables, Port.Math.