A characterization of the Filippov convention

The Filippov convention is widely used in the literature to define vector fields on a discontinuity set of piecewise-continuous vector fields. The aim of this paper is to give a sufficient and necessary condition for an interpolation scheme of piecewise-continuous vector fields to coincide with the Filippov convention. That is, we show that a map from a space of piecewise-continuous vector fields with two components to the space of vector fields coincides with the Filippov convention where the latter can be applied, if it is sufficiently well-behaved as a generalization of continuous vector fields.