The generic differentiability of convex-concave functions: Characterization

As established by R T. Rockafellar, real valued convex-concave functions are generically differentiable. It this paper we shall show that for a convex-concave function defined on an open convex set $C\times D,$ there exist dense subsets $\mathcal{N}$ of $C$ and $\mathcal{M}$ of $D$ such that the partial derivative with respect to the first variable (resp. second variable) exists on $\mathcal{N}\times D$ (resp. $C\times \mathcal{M}$) and therefore the function is differentiable on $\mathcal{N}\times \mathcal{M}$. This is an interesting property of convex-concave functions and it does not hold for convex-convex functions. As an immediate application we recover the generic single-valuedness of monotone operators.