Graphs of Gâteaux derivatives are w*-connected.

Abstract

We show that if $(X, \|. \|)$ is a separable Banach space, $\Omega \subset X$ is open, connected and $f: \Omega \to \mathbb{R}$ is an everywhere Gâteaux differentiable Lipschitz continuous function, then the graph of the derivative of $f$ is connected in $(\Omega,\| . \|)\times(X^\star , w^\star)$.