Basic elements of nonsmooth analysis and on the continuities of multivalued mappings

In this work Dini, Hadamard and Clarke directional derivatives are defined and according to these the concepts of subdifferential and Clarke subdifferential are given. Then the concepts of tangent cones and normal cones of a set are defined and some relationships between these cones and subdifferential, Clarke subdifferential and directional derivatives are studied. In addition, upper and lower semicontinuities, Lipschitsz, pseudo Lipschitz and pseudo Hölder continuities of multivalued mappings are defined, and some important proporties of them are expressed. At the same time, relations between continuity of multivalued mappings and the continuities of support, distance functions and marginal mappings which are defined by multivalued mappings are studied. At the end, concept of convex processes is defined and some proporties of it are given.