A class of one-dimensional classical lattice gas models with translationally invariant, stable and strongly tempered many-body interactions, is constructed for which the limiting thermodynamic pressure is a discontinuous function of the density (at fixed temperature). This demonstrates that an apparently “mild” restriction on the potential (“supersummability”) employed by Griffiths and Ruelle in proving the continuity of the pressure in lattice systems, plays, in fact, a crucial role.