AbstractRate independent operators naturally arise in the mathematical analysis of hysteresis. Among rate independent operators, the locally monotone ones are those better suited for the study of PDE's with hysteresis. We prove that a rate independent operator R:Lip(0,T) →BV(0,T) ∩C(0,T) which is locally monotone and continuous with respect to the strict topology ofBVadmits a unique continuous extension R̃:BV(0,T) →BV(0,T). This general result applies to several concrete hysteresis operators. For many of these operators the existence of a continuous extension was previously known at most to the spaceBV(0,T) ∩C(0,T). (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)