In the theory of optimization several types of piecewise differentiable functions occur in a quite natural way. As a typical example
for such nondifferentiable functions we mention the finite max-min combinations of differentiable functions. A more general class are the quasidifferentiable functions, which are investigated in detail by V. F. Demyanov and A. M. Rubinov (see for instance [1]).
The directional derivatives of these functions can be represented as a difference of two sublinear functions. Since a sublinear function is uniquely described by its subdifferential in the origin, there exists a natural correspondence between the directional derivatives and the set of pairs of convex compact sets. In this paper we give a comprehensive representation of the results in [5], [6] and [7]. Moreover we show that there exists a natural definition for the difference between pairs of convex compact sets.