Convexity, concavity, super-additivity, and sub-additivity of cost function without fixed cost

With zero fixed cost, convexity of a cost function implies super-additivity, and concavity of a cost function implies sub-additivity. But converse relations do not hold. However, in addition to the zero fixed cost condition we put the following assumption. (1) If a cost function is convex in some interval, it is convex throughout the domain. (2) If a cost function is concave in some interval, it is concave throughout the domain. Then, super-additivity implies convexity and sub-additivity implies concavity. Subsequently, super-additivity and convexity are equivalent, and sub-additivity and concavity are equivalent.