Cone-monotone functions: Differentiability and continuity

We provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone K with non-empty interior. We also show that the set of nowhere K-monotone functions has a sigma-porous complement in the space of continuous functions endowed with the uniform metric.