Differentiability of Approximately Convex, Semiconcave and Strongly Paraconvex Functions

It is shown that continuous approximately convex, semiconcave and strongly $\alpha(\cdot)$-paraconvex functions on Banach spaces have almost all (but not all) known first order differentiability properties of continuous convex functions. The main results easily follow from known (or essentially known) results on single-valuedness and continuity of submonotone operators.