Can a non-Lipschitz function operate non-trivially on a Banach space of functions? (Q1328323)

Let \(E\) be a subspace of \(C(K)\) (\(K\) a compact set) which separates points and contains the constant functions, and let \(\varphi\) be some real function. The author studies the following problem: Characterize those \(\varphi\) such that the corresponding autonomous Nemytskij operator \(u\mapsto \varphi\circ u\) maps \(E\) into itself. For example, for a non- affine function \(\varphi\) this is possible only for \(E= C(K)\).