arXiv:1509.02965v1 [math.CA] 9 Sep 2015 A note on a global invertibility of locally lipschitz functions on R n M. Galewski and M. R˘ adulescu December 1, 2018 Abstract We provide sufficient conditions for a locally lipschitz mapping f : R n → R n to be invertible . We use classical local invertibility condi- tions together with the non-smooth critical point theory. 1 Introduction In this note using non-smooth critical point theory together with classical local invertibility results we consider locally invertible locally Lipschitz func- tions f : R n → R n . We provide conditions for global invertibility of f . This note aims at answering the question posed on p. 20 of [2] which suggests that results from [3] should have their Lipschitz counterpart. The difficulties which arise in the non-smooth setting are described to the end of this sec- tion and although we follow the pattern introduced in [3], i.e. we make local invertibility a global one, we do not have sufficient tools to do this directly. The local results we base on are as follows Lemma 1. [1]Let D be an open subset of R n . If f : D → R n satisfies a Lipschitz condition in some neighbourhood of x 0 and ∂f (x 0 ) ⊆ R n (where ∂f (x 0 ) denotes the collection of all generalized directional derivative of f at the point x ∈ R n in the sense of Clarke) is of maximal rank, then there exist neighbourhoods U ⊂ D of x 0 and V of f (x 0 ) and a Lipschitz function g : V → R n such that i ) for every u ∈ U , g (f (u)) = u, and ii) for every v ∈ V , f (g (v)) = v. 1