  For a probability P in $R^d$ its center outward distribution function
$F_{\pm}$, introduced in Chernozhukov et al. (2017) and Hallin et al. (2021),
is a new and successful concept of multivariate distribution function based on
mass transportation theory. This work proves, for a probability P with density
locally bounded away from zero and infinity in its support, the continuity of
the center-outward map on the interior of the support of P and the continuity
of its inverse, the quantile, $Q_{\pm}$. This relaxes the convexity assumption
in del Barrio et al. (2020). Some important consequences of this continuity are
Glivenko-Cantelli type theorems and characterisation of weak convergence by the
stability of the center-outward map.
