  Let $f: M\rightarrow M$ be a continuous map on a compact metric space $M$
equipped with a fixed metric $d$, and let $\tau$ be the topology on $M$ induced
by $d$. First, we will establish some fundamental properties of the mean
Hausdorff dimension. Furthermore, it is important to note that the metric mean
dimension and mean Hausdorff dimension depend on the metric chosen for $M$. In
this work, we will prove that, for a fixed dynamical system $f:M\rightarrow M$,
the functions $\text{mdim}_{\text{M}}(M, f):M(\tau)\rightarrow \mathbb{R}\cup
\{\infty\}$ and $\text{mdim}_{\text{H}}(M, f):M(\tau)\rightarrow \mathbb{R}\cup
\{\infty\}$ are not continuous. Here, $ \text{mdim}_{\text{M}}(M, f)(\rho)=
\text{mdim}_{\text{M}}(M,\rho, f)$ and $ \text{mdim}_{\text{H}}(M, f)(\rho)=
\text{mdim}_{\text{H}}(M,\rho, f)$ represent, respectively, the metric mean
dimension and the mean Hausdorff dimension of $f$ with respect to $\rho\in
M(\tau)$ and $M(\tau)$ is the set consisting of all equivalent metrics to $d$
on $M$. Furthermore, we will present examples of certain classes of metrics for
which the metric mean dimension is a continuous function.
