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On a metric space equipped with a doubling measure supporting a Poincar\'einequality, we show that given a BV function, discarding a set of small$1$-capacity makes the function continuous outside its jump set and``one-sidedly" continuous in its jump set. We show that such a propertyimplies, in particular, that the measure theoretic boundary of a set of finiteperimeter separates the measure theoretic interior of the set from its measuretheoretic exterior, both in the sense of the subspace topology outside sets ofsmall $1$-capacity, and in the sense of $1$-almost every curve.

Fine properties and a notion of quasicontinuity for BV functions on metric spaces