Directional mean dimension and continuum-wise expansive $mathbb{Z}^k$-actions

We study directional mean dimension of $mathbb{Z}^k$-actions (where $k$ is a positive integer). On the one hand, we show that there is a $mathbb{Z}^2$-action whose directional mean dimension (considered as a $[0,+infty]$-valued function on the torus) is not continuous. On the other hand, we prove that if a $mathbb{Z}^k$-action is continuum-wise expansive, then the values of its $(k-1)$-dimensional directional mean dimension are bounded. This is a generalization (with a view towards Meyerovitch and Tsukamotos theorem on mean dimension and expansive multiparameter actions) of a classical result due to Ma~ne: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional.