Let f be a regular, locally Lipschitz real-valued function defined on an open convex subset of a normed space. We show that any unit direction u, the upper second-order derivative D+2f(·; u, 0) has the same lower bounds as the lower second-order derivatives D−2f(·; u, 0). Consequently, one can characterize the convexity of f in terms of these derivatives. We also obtain the corresponding results in terms of Chaney's second-order directional derivatives.
