Continuity of p-variation in the Vietoris topology

Abstract

Some generalizations of two known results regarding the role of points of varying monotonicity in computing p-variation and two special variants of p-variation are given. Further, it is shown that if f is a continuous function of bounded p-variation, then v p ( f , A ) —the p-variation of f on a closed subset A—is a continuous function of A in the Vietoris topology. A decomposition of a modified Love–Young starred p-variation v p ∗ is proven, yielding a new description of the Wiener class W p ∗ . A notion of equidiscontinuous functions is introduced and used to provide another characterization of functions of class W p ∗ .