On locally isotone rate independent operators

This note concerns rate independent operators R:Lip(0,T)⟶BV(0,T)∩C(0,T) that are continuous with respect to the strict metric of BVBV. In a recent paper we proved that R can be continuously extended to all of BV(0,T)BV(0,T) provided it is locally isotone, a slightly weaker condition than the local monotonicity well known in hysteresis. In this work we prove that local isotony is also a necessary condition, thereby giving a characterization of rate independent operators R that admit a continuous extension to BVBV.