Since continuity of partial derivatives implies differentiability of the function, formula 108 is indeed differentiable. The isomorphisms in this case are bijections with the chosen degree of differentiability. The hard part of the theorem is the existence and differentiability of formula 22. Stronger statement than differentiability can be made regarding the resolvent map. Continuous Gateaux differentiability may be defined in two inequivalent ways. A function's being odd or even does not imply differentiability, or even continuity. The assumption of differentiability can be weakened to continuity and one-sided differentiability of "f". Jaynes reproduces the shorter proof by Cox in which differentiability is assumed. The Sobolev spaces formula 32 combine the concepts of weak differentiability and Lebesgue norms. It can be extended to much wider classes of functions satisfying mild differentiability conditions. The relationship between real differentiability and complex differentiability is the following. Consequently, complex differentiability has much stronger implications than real differentiability. All functions are assumed to be of differentiability class "C" unless otherwise noted. Differentiability class is a classification of functions according to the properties of their derivatives. One can classify functions by their differentiability class.