Remarks and improvement regarding the theorem of continuous dependence of solutions of F.D.E., using some new results for continuous convergences

We improve the theorem on continuous dependence of solutions of functional differential equations (see J. Hale, Functional differential equations, theorem 5.1), using some new results on continuous convergences. Namely, we prove this theorem without the assumption of uniform boundeness for the sequence (f_k). Also we restore some inconsistencies regarding the proof of this theorem.