arguments In infinite dimensions, any discontinuous linear functional on is Gateaux differentiable, but its Gateaux differential at is linear but not continuous. Relation with the Fréchet derivative If is Fréchet differentiable, then it is also Gateaux differentiable, and its Fréchet and Gateaux derivatives agree. The converse is clearly not true, since the Gateaux derivative may fail to be linear or continuous. In fact, it is even possible for the Gateaux derivative to be linear and continuous but for the Fréchet derivative to fail to exist. Nevertheless, for functions from a Banach space to another complex Banach space the Gateaux derivative (where the limit is taken over complex tending to zero as in the definition of complex differentiability) is automatically linear, a theorem of . Furthermore, if is (complex) Gateaux differentiable at each with derivative then is Fréchet differentiable on with Fréchet derivative . This is analogous to the result from basic complex analysis that a function is analytic if it is complex differentiable in an open set, and is a fundamental result in the study of infinite dimensional holomorphy. Continuous differentiability Continuous Gateaux differentiability may be defined in two inequivalent ways. Suppose that is Gateaux differentiable at each point of the open set One notion of continuous differentiability in requires that the mapping on the product space be continuous. Linearity need not be assumed: if and are Fréchet spaces, then is automatically bounded and linear for all . A stronger notion of continuous differentiability requires that be a continuous mapping from to the space of continuous linear functions from to Note that this already presupposes the linearity of As a matter of technical convenience, this latter notion of continuous differentiability is typical (but not universal) when the spaces and are Banach, since is also Banach and standard results from functional analysis can then be employed. The former is the more common definition in areas of nonlinear analysis where the function spaces involved are not necessarily Banach spaces. For instance, differentiation in Fréchet spaces has applications such as the Nash–Moser inverse function theorem in which the function spaces of interest often consist of smooth functions on a manifold. Higher derivatives Whereas higher order Fréchet derivatives are naturally defined as multilinear functions by iteration, using the isomorphisms higher order Gateaux derivative cannot be defined in this way. Instead the th order Gateaux derivative of a function in the direction is defined by