I Generalized Characteristics of First-Order PDEโ{128}{153}s -- II Cauchy Problems for Hamilton-Jacobi Equations -- III Differential Games -- IV Boundary-Value Problems for First-Order PDEโ{128}{153}s -- A1 Justification of the Classical Method of Characteristics -- A2 Multifunctions -- A3 Semicontinuous Functions -- A4 Convex Functions -- A5 Contingent Tangent Cones, Directional Derivatives, Subdifferentials -- A6 On a Property of Subdifferentials -- A7 Differential Inclusions -- A8 Criteria for Weak Invariance