An alternative approach to Fréchet derivatives

We discuss an alternative approach to Frechet derivatives on Banach spaces inspired by a characterisation of derivatives due to Caratheodory. The approach allows many questions of differentiability to be reduced to questions of continuity. We demonstrate how that simplifies the theory of differentiation, including the rules of differentiation and the Schwarz lemma on the symmetry of second-order derivatives. We also provide a short proof of the differentiable dependence of fixed points in the Banach fixed point theorem.