Duality between differentiability and rotundity for convex functions

We generalize a theorem of Asplund and Rockafellar concerning the duality between the differentiability of a convex function with respect to a system of sets $\mathscript A$ and the $\mathscript A$ rotundity of the conjugate convex function to the case of differentiabilities and rotundities defined by pseudotopologies of a broad class, including pseudotopologies of convergence on systems of filters and their bulblike modifications.