A Mayer problem of optimal control, whose dynamic constraint is given by a convex-valued differential inclusion, is considered. Both state and endpoint constraints are involved. Necessary conditions are proved incorporating the Hamiltonian inclusion, the Euler–Lagrange inclusion, and the Weierstrass–Pontryagin maximum condition. These results weaken the hypotheses and strengthen the conclusions of earlier works. Their main focus is to allow the admissible velocity sets to be unbounded, provided they satisfy a certain continuity hypothesis. They also sharpen the assertion of the Euler–Lagrange inclusion by replacing Clarke’s subgradient of the essential Lagrangian with a subset formed by partial convexification of limiting subgradients. In cases where the velocity sets are compact, the traditional Lipschitz condition implies the continuity hypothesis mentioned above, the assumption of “integrable boundedness” is shown to be superfluous, and this refinement of the Euler–Lagrange inclusion remains a strict improvement on previous forms of this condition.