We define the speed measure ν for mappings γ: I X from an interval to a metric space that are locally of bounded variation. We characterize continuity and absolute continuity of γ in terms of ν and identify the Radon-Nikodým derivative of ν with respect to Lebesgue measure as the metric speed of γ. In doing so we prove an extension of the Banach-Zaretsky theorem.
Boldt et al. (Thu,) studied this question.