The authors demonstrated in the first part of this paper [ibid. 23, 566- 583 (1985; Zbl 0569.49019)] the connections between the notion of viscosity sub- and super-solutions of first-order, dynamic programming PDE and the optimality principle of dynamic programming, as well as the directional derivatives of viscosity solutions of the above equations at an arbitrary point. The present note contains a remark and a counterexample which complement the results cited.