Let x =f(x, μ, v) be the kinematic equation of a two-person, zero-sum differential game where (μ, v) is of class C(m) on the playing space. It is shown that, whenever f(x, μ, v) is not tangent to the terminal surface ϕ there exists a neighbourhood of ϕ in which every point y can he uniquely represented as γ(τ,y0) where γ is a solution to the kinematic equations, y0is in ϕif, and τ is the time until the path originating at y meets ya. This representation yields two functions W(y ; μ, v), the pay-off at y resulting from (μ, v), and T(y ;μ, v), the terminating time from y. It is shown that these functions are of class C(m); and their gradients satisfy the same adjoint equation but with different initial transversality conditions, even when (μ, v) are not optimal. These results are then extended to a carefully defined class of discontinuous strategies, those containing a ’ transition surface ’ or ’ discontinuity manifold ’. Lastly, those results arc used to give a condition for closed-loop optimality in optimal control problems and for semipermoability in differential games. These results are easily generalizablo to n-player games.