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$BV[0,1]$
The main goal of this paper is to give the answer to one of the main problems of the theory of nonautonomous superposition operators acting in the space of functions of bounded variation in the sense of Jordan. Namely, we prove that if the superposition operator maps the space 
$BV[0,1]$ into itself, then it is automatically locally bounded, provided its generator is a locally bounded function.
