On the sign of the static dielectric function at T=0

The question of the allowed signs for the static dielectric function for exactly homogeneous ground states in many body systems is further analyzed. The discussion is restricted to zero temperature situation. Firstly, it is argued that the positivity is equivalent to the requirement that the work needed for to adiabatically connect a static charge density, be positive. In particular the rule seems to be a fully appropriate stability condition if the system interacts with a freely moving but coulomb interacting jellium. This situation is common in thermodynamical equilibrium discussions, in which the system can be stable or not in dependence of the relaxation of some external constraints. Further, an argument based in a quantized electromagnetic field treatment is also given. It requires the positivity of the dielectric function as an intrinsic stability condition for the charged system independently of the jellium dynamics.Possible inmplications for the more complex finite temperature situation are commented.