The Failure of Rolle’s Theorem in Infinite-dimensional Banach Spaces

We prove the following new characterization of C (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a C smooth (Lipschitz) bump function if and only if it has another C smooth (Lipschitz) bump function f such that its derivative does not vanish at any point 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. The “twisted tube” method we use in the proof is interesting in itself, as it provides other useful characterizations of C smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer’s fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces.