Banach–Zaretsky theorem for compactly absolutely continuous mappings

For mappings of an interval into locally convex spaces, convex and compact convex analogs of absolute continuity, bounded variation, and the Luzin $N$-property are introduced and studied. We prove that, in the general case, a convex analog of the Banach–Zaretsky criteria can be “split” into sufficient and necessary conditions. However, in the Fréchet-space case, we have an exact compact analog of the criteria.