Study on differentiability problems of interval-valued functions

In this paper, we give the concepts of H-directional differentiability and D-directional differentiability of interval-valued functions. Then we discuss the properties of H-directional differentiable interval-valued functions and D-directional differentiable interval-valued functions. The necessary and sufficient conditions for the H-directional differentiability are given together with the sufficient conditions and the necessary and sufficient conditions for D-directional differentiability of interval-valued functions. Then we discuss the relationship between the two directional differentiability and prove these directional differentiability can be equivalent under a certain conditions.