The Graves Theorem Revisited

In 1950 Graves proved the following theorem: If the function f from a Banach space X into a Banach space Y is strictly differentiable at $x_0$ and the strict derivative $\nabla f(x_0)$ is onto, then f is open with linear rate around $x_0$. Under fairly general assumptions, the latter property is equivalent to either the metric regularity or to the Aubin property of the inverse. In this paper, we prove that the Graves theorem is a consequence of the following general result: the openness with linear rate of a locally closed set-valued map F around a point $(x_0, y_0)$ of its graph is invariant with respect to a perturbation of the form f + F provided that the strict derivative of f at $x_0$ is zero.