Let $\Gamma(x)$ be a correspondence (i.e. a set-valued function) between two Euclidean spaces which is continuous (i.e. both lower- and upper-hemicontinuous). If $y$ is a point in the interior of $\Gamma(x_0)$ it seems plausible from drawing graphs that there should be an open set $U$ containing $x_0$ such that $y$ is in the interior of $\Gamma(x)$ for all $x \in U$.
Is this a correct theorem? I would appreciate some references or other pointers.