The continuity of a convex vector function on relative interior points of its domain is studied. As a corollary of this we can see that a convex vector function is Lipschitz near any relative interior point of its domain. A new concept of subdifferential of a convex function is introduced and some its properties similar to those in the scalar case are shown. The inclusive relations between generalized Jacobian and subdifferential, the convexcity of a vector function and the monotonicity of its subdifferential are also established. Further, some neccessary and sufficient conditions for the existence of efficient solutions of vector optimization problems are also proved.