This work proposes a general framework for analyzing the behavior at its extrema of an extended real-valued function assumed neither convex nor differentiable and for which the classical Fermat rules of optimality do not apply. The tools used for building this frame are the notions of sup-subdifferential, recently introduced by two of the authors together with A. Kruger, and partial sup-subdifferentials. The sup-subdifferential is always a \textit{nonempty} enlargement of the Moreau-Rockafellar subdifferential from convex optimization. It satisfies most of the fundamental properties of the Moreau-Rockafellar subdifferential and enjoys certain calculus rules. The partial sup-subdifferentials are obtained by breaking down the sup-subdifferential into one-dimensional components through basis elements and play the same role as the partial derivatives in the Fermat optimality rules.