A note on the composition of regular functions (Q2448602)

The authors prove that a nonlinear autonomous composition operator maps a Banach space of regular functions \(f: [ a,b] \to {\mathbb R}\) into itself if and only if the generating function is continuous on \({\mathbb R}\). A sufficient condition in the case where an operator is nonautonomous is also given. Moreover, they construct an example showing that, if a generating function is regular, then the corresponding autonomous composition operator need not map the class of regular functions into itself.