A Note on Clarke’s Generalized Jacobian for the Inverse of Bi-Lipschitz Maps

Clarke’s inverse function theorem for Lipschitz mappings states that a bi-Lipschitz mapping f is locally invertible about a point $$x_0$$ x 0 if the generalized Jacobian $$\partial f(x_0)$$ ∂ f ( x 0 ) does not contain singular matrices. It is shown that under these assumptions the generalized Jacobian of the inverse mapping at $$f(x_0)$$ f ( x 0 ) is the convex hull of the set of matrices that can be obtained as limits of sequences $$J_f(x_k)^{-1}$$ J f ( x k ) - 1 with f differentiable in $$x_k$$ x k and $$x_k$$ x k converging to $$x_0$$ x 0 . This identity holds as well if f is assumed to be locally bi-Lipschitz at $$x_0$$ x 0 .