  We discuss some aspects of the continuum limit of some lattice models, in
particular the $2D$ $O(N)$ models. The continuum limit is taken either in an
infinite volume or in a box whose size is a fixed fraction of the infinite
volume correlation length. We point out that in this limit the fluctuations of
the lattice variables must be $O(1)$ and thus restore the symmetry which may
have been broken by the boundary conditions (b.c.). This is true in particular
for the so-called super-instanton b.c. introduced earlier by us. This
observation leads to a criterion to assess how close a certain lattice
simulation is to the continuum limit and can be applied to uncover the true
lattice artefacts, present even in the so-called 'perfect actions'. It also
shows that David's recent claim that super-instanton b.c. require a different
renormalization must either be incorrect or an artefact of perturbation theory.
