differentiability

Now, the study of differentiability is a central concern in nonlinear functional analysis.

The differentiability and continuity of these functions are then established from the series definitions alone.

Higher order differentiability classes correspond to the existence of more derivatives.

Roughly speaking, differentiability puts a bound on how fast the curve can turn.

In particular, continuous differentiability of f need not be assumed .

Stronger statement than differentiability can be made regarding the resolvent map.

In fact in general, the notion of differentiability is not defined on a metric space.

The next step is to use the definition of differentiability of f at g(a).

Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability).

They are defined as inner products on the tangent space with an appropriate differentiability requirement.

The isomorphisms in this case are bijections with the chosen degree of differentiability.

The relationship between real differentiability and complex differentiability is the following.

It can be extended to much wider classes of functions satisfying mild differentiability conditions.

Differentiability class is a classification of functions according to the properties of their derivatives.

Continuous Gâteaux differentiability may be defined in two inequivalent ways.

These criteria of differentiability can be applied to the transition functions of a differential structure.

The definition of differentiability depends on the choice of chart at p; in general there will be many available charts.

A "quadratic approximation" (to the differentiability requirement) is given by:

The definition of strict differentiability avoids this problem by imposing a condition directly on the difference quotients.

The problem with the above approach is that it relies on the differentiability of the objective function and on concavity.

A function's being odd or even does not imply differentiability, or even continuity.

It is also related to a completely general definition of differentiability given by Carathéodory .

Differentiability of Lipschitz functions on metric measure spaces.

In this example one can easily see that pointwise convergence does not preserve differentiability or continuity.

Assume differentiability and that is the solution at , then we have from the multivariate chain rule: