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

On a metric space equipped with a doubling measure supporting a Poincar'e inequality, 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 property implies, in particular, that the measure theoretic boundary of a set of finite perimeter separates the measure theoretic interior of the set from its measure theoretic exterior, both in the sense of the subspace topology outside sets of small $1$-capacity, and in the sense of $1$-almost every curve.