On diffeomorphisms deleting weakly compacta in Banach spaces

Rolle's theorem is either false or trivial in infinite-dimensional Banach spaces

We prove the following new characterization of $C^p$ (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space $X$ has a $C^p$ smooth (Lipschitz) bump function if and only if it has another $C^p$ smooth (Lipschitz) bump function $f$ such that $f'(x)\neq 0$ for every point $x$ in the interior of the support of $f$ (that is, $f$ does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. As a by-product of the proof of this result we also obtain other useful characterizations of $C^p$ smoothness related to the existence o…