### A characterization of ${C}^{1,1}$ functions via lower directional derivatives

The notion of $\tilde{\ell}$-stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of $\tilde{\ell}$-stable functions coincides with the class of C${}^{1,1}$ functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp. 383–387.