The conditions for a discontinuous function to be identical with the value function of a game in a time-optimal problem

A differential game is considered in which the time until a point reaches a target set is the pay functional. The sufficient conditions for the given discontinuous function to be identical with the value function of the game are obtained. The conditions are formulated in terms of the classical concepts of u-   and vv-stable functions but it is additionally required that the so-called condition of the proper contractibility of the closed sets of the level of the function being tested is satisfied. An example is presented which shows that the condition of proper contractibility is not excessive.