This thesis focuses on the study of differentiated functions defined on subsets of Banach spaces, in particular we study the convex and continuous functions and more specifically the norm. It shows the intimate relationship between different types of differentiated (Fréchet, Gâteaux strongly subdiferenciable quite soft, ...) and the topological structure of Banach spaces which are defined functions (Asplund spaces, separability, the dual space has no subspaces normantes own rules rough ...) We conclude the thesis with the study of the relationship between the topological properties anteriormetne these subsets and the immersion of the weak-* homeomorfos Cantor ternary set in the field dual unit.