Jacobians and branch points of real analytic open maps

Introduction The main object of this paper is to prove the following result: Theorem 1 The Jacobian of a real analytic open map f : R → R does not change sign. One of the referees kindly pointed out that the special case of polynomial maps was proved by Gamboa and Ronga [3]: Theorem 2 (Gamboa and Ronga) A polynomial map in R is open if and only if point inverses are finite and the Jacobian does not change sign. The proof of Theorem 1 is very similar to methods in [3], which are easily adapted to analytic maps; but as Theorem 1 does not seem to be known, a direct proof may be useful. f : R → R denotes a (real) analytic map in Euclidean n-space. We always assume f is open, that is, f maps open sets onto open sets. Denote the Jacobian matrix of f at p ∈ R by dfp = [