The Local Rotation Set is an Interval

Let $Homeo\_0 (R 2 ; 0)$ be the set of all homeomorphisms of the plane isotopic to the identity and which fix 0. Recently in the article entitled "L'ensemble de rotation local autour d'un point fixe" Fr{\'e}d{\'e}ric Le Roux gave the definition of the local rotation set of an isotopy in $Homeo\_0 (R 2 ; 0)$ from the identity to a homeomorphism f and he asked if this set is always an interval. In this article we give a positive answers to this question and to the analogous question in the case of the open annulus.