A selection theorem for essentially regulated functions of several variables

Helly's selection theorem provides a criterion for compactness of sets of single-variable functions with bounded pointwise variation. Fraňková has given a proper extension of Helly's theorem to the setting of single-variable regulated functions. We show how a similar approach yields a natural notion of regulated functions of several variables, together with an extension of the standard compactness criterion for multi-variable functions of bounded variation.