Two-person zero-sum continuous games with concave-convex payoffs are considered. A necessary and sufficient condition for the existence of a solution to the game is proved. As a sufficient condition this generalizes a well-known theorem of von Neumann, but as a necessary condition it is new. Here the convexity notion is extended by the assumption of the existency only of a point \(x_{\alpha}\) instead of the traditional \(x_{\alpha}=\alpha x^ 1+(1-\alpha)x^ 2\). The proved criterion is the formulation of the upper semicontinuity of a special functional in the point zero.