  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
Tsukamoto's theorem on mean dimension and expansive multiparameter actions) of
a classical result due to Ma\~n\'e: Any compact metrizable space admitting an
expansive homeomorphism (with respect to a compatible metric) is
finite-dimensional.
