An Open Mapping Theorem using unbounded Generalized Jacobians∗

In this paper, three key theorems (the open mapping theorem, the inverse function theorem, and the implicit function theorem) for continuously differentiable maps are shown to hold for nonsmooth continuous maps which are not necessarily Lipschitz continuous. The significance of these extensions is that they are given using generalized Jacobians, called approximate Jacobians. The approximate Jacobian which replaces the non-existent Jacobian matrix at the points of nondifferentiability for (not necessarily Lipschitzian) nonsmooth continuous maps by an unbounded set of matrices, enjoy rich and often exact calculus, and produce sharp results for Lipschitzian problems. The main tools are a generalized mean value theorem for continuous maps and the recession cones of unbounded approximate Jacobians. A general chain rule formula for these approximate Jacobians play a crucial role in the extensions.