Let $n\in\mathbb{N}$
Is the volume functional continuous on the set of isometry classes of compact riemannian $n$-manifolds with volume $\geq \varepsilon$\
(with respect to Gromov--Hausdorff distance)?


Without the volume bound,
a collapsing torus gives a counterexample. 
But it seems that this is the only singularity.
There are a number of weird results on the semi-continuity of the volume functional.
For instance,
every metric on $S^3$ is the limit of a metric with volume converging to $0$.
I don't know what happens to the curvature of these metrics.



Is the volume functional contiunuous for compact manifolds with lower bounds on volume?
I assume you talk about continuity in Gromov-Hausdorff topology. 


In general, volume is not continuous, as easy examples show (see e.g. Colding's paper "Large manifolds with positive Ricci curvature", examples 1-2 which have a lower volume bound).


On the other hand, for manifolds with a lower Ricci curvature bound, Colding shows the for each $r$ the volume of $r$-balls is continuous in Gromov-Hausdorff topology; see 
his paper "Ricci curvature and volume convergence".


