  The determination of the resolution of cosmological N-body simulations, i.e.,
the range of scales in which quantities measured in them represent accurately
the continuum limit, is an important open question. We address it here using
scale-free models, for which self-similarity provides a powerful tool to
control resolution. Such models also provide a robust testing ground for the
so-called stable clustering approximation, which gives simple predictions for
them. Studying large N-body simulations of such models with different force
smoothing, we find that these two issues are in fact very closely related: our
conclusion is that the accuracy of two point statistics in the non-linear
regime starts to degrade strongly around the scale at which their behaviour
deviates from that predicted by the stable clustering hypothesis. Physically
the association of the two scales is in fact simple to understand: stable
clustering fails to be a good approximation when there are strong interactions
of structures (in particular merging) and it is precisely such non-linear
processes which are sensitive to fluctuations at the smaller scales affected by
discretisation. Resolution may be further degraded if the short distance
gravitational smoothing scale is larger than the scale to which stable
clustering can propagate. We examine in detail the very different conclusions
of studies by Smith et al. (2003) and Widrow et al. (2009} and find that the
strong deviations from stable clustering reported by these works are the
results of over-optimistic assumptions about scales resolved accurately by the
measured power spectra, and the reliance on Fourier space analysis. We
emphasise the much poorer resolution obtained with the power spectrum compared
to the two point correlation function..
