A method of discrete approximation of the reachable set of a differential inclusion \(x'\in F(x)\), \(x(0)=\xi\), with \(F\) Lipschitzian, is proposed. (Similar results have been given by A. I. Panasyuk and V. I. Panasyuk [Math. Notes 27, 213-218 (1980); translation from Mat. Zametki 27, 429-437 (1980; Zbl 0443.49031)].) The result is applied to the problem of characterization of the value function of the optimal control problem of Mayer.