On the putative essential discreteness of q-generalized entropies

It has been argued in Abe (2010), entitled Essential discreteness in generalized thermostatistics with non-logarithmic entropy, that ”continuous Hamiltonian systems with long-range interactions and the so-called q-Gaussian momentum distributions are seen to be outside the scope of non-extensive statistical mechanics”. The arguments are clever and appealing. We show here that, however, some mathematical subtleties render them unconvincing.