On the continuity of superposition operators in the space of functions of bounded variation

In the paper we present results on the continuity of nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan. It is shown that the continuity of an autonomous superposition operator is automatically guaranteed if the acting condition is met. We also give a simple proof of the fact that a nonautonomous superposition operator generated by a continuously differentiable function is uniformly continuous on bounded sets. Moreover, we present necessary and sufficient conditions for the continuity of a superposition operator (autonomous or nonautonomous) in a general setting. Thus, we give the answers to two basic open problems mentioned in the monograph (Appell et al. in Bounded variation and around, series in nonlinear analysis and application, De Gruyter, Berlin, 2014).