Analysis MOC

Differentiability

A map is -differentiable at a point iff it has an -th derivative at that point, and thus all derivatives up to . anal Moreover is called -differentiable if it is differentiable at every . is the set of all -differentiable functions with a continuous th derivative, and is called a differentiability class, and . In particular,

• is the class of all continuous functions;

• of analytic functions; and

• of infinitely differentiable functions¹.

Generalizations

Complex functions

In complex analysis all differentiable functions are analytic and infinitely differentiable. Such a function is called holomorphic.

Open subsets of real coördinate space

Differentiability generalizes naturally to higher dimensional Real coördinate space (and open subsets thereof). A function is iff it has all -th order partial derivatives.

Arbitrary subsets of real coördinate space

Let be inhabited. A function is iff every has an open neighbourhood with a extension such that for all . diff

By considering real submanifolds, this yields the notion of differentiability for maps between such manifolds.

Map between manifolds

Let be a map between manifolds of dimension and respectively. The class is only well defined if and are differentiable manifolds. Let . is called -differentiable at iff there exists a chart containing and containing such that is -differentiable at . diff is called iff it is -differentiable everywhere.

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Footnotes

1. Often called smooth, however the exact meaning of this term varies between authors, e.g. sometimes functions are referred to as smooth. ↩