  This article provides a definition of a subdifferential for continuous
functions based on homological considerations. We show that it satisfies all
the requirement for a good notion of subdifferential. Moreover, we prove
sublinearity, a Leibniz formula and an approximation result. This work fits in
the framework of microlocal analysis of sheaves for C^0 symplectic problems and
application to Aubry-Mather theory.
