A notion of fine continuity for BV functions on metric spaces

TL;DR

This paper introduces a new notion of fine continuity for BV functions on metric spaces with doubling measures and Poincaré inequalities, showing they are continuous in a 1-fine topology at almost every point.

Contribution

It establishes a novel concept of fine continuity for BV functions in metric spaces and proves their continuity at almost every point in this refined topology.

Findings

BV functions are 1-finely continuous at almost every point

The result applies to metric spaces with doubling measures and Poincaré inequalities

Introduces a new notion of fine continuity for BV functions

Abstract

In the setting of a metric space equipped with a doubling measure supporting a Poincar\'e inequality, we show that BV functions are, in the sense of multiple limits, continuous with respect to a 1-fine topology, at almost every point with respect to the codimension 1 Hausdorff measure.