Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (differentiability class) it has over its domain.

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

It is a measure of the highest order of derivative that exists and is continuous for a function.

is an example of a function that is differentiable but not locally Lipschitz continuous.

is analytic, and hence falls into the class Cω (where ω is the smallest transfinite ordinal).

is a Fréchet vector space, with the countable family of seminorms

varies over an increasing sequence of compact sets whose union is

The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of partial differential equations, it can sometimes be more fruitful to work instead with the Sobolev spaces.

The terms parametric continuity (Ck) and geometric continuity (Gn) were introduced by Brian Barsky, to show that the smoothness of a curve could be measured by removing restrictions on the speed, with which the parameter traces out the curve.

As a practical application of this concept, a curve describing the motion of an object with a parameter of time must have C1 continuity and its first derivative is differentiable—for the object to have finite acceleration.

For smoother motion, such as that of a camera's path while making a film, higher orders of parametric continuity are required.

[8][9] A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.

continuity at the point where they meet if they satisfy equations known as Beta-constraints.

continuity to appear smooth, for good aesthetics, such as those aspired to in architecture and sports car design, higher levels of geometric continuity are required.

[citation needed] A rounded rectangle (with ninety degree circular arcs at the four corners) has

The same is true for a rounded cube, with octants of a sphere at its corners and quarter-cylinders along its edges.

continuity is required, then cubic splines are typically chosen; these curves are frequently used in industrial design.

Simple examples of functions that are smooth but not analytic at any point can be made by means of Fourier series; another example is the Fabius function.

Furthermore, for every open subset A of the real line, there exist smooth functions that are analytic on A and nowhere else.

[citation needed] It is useful to compare the situation to that of the ubiquity of transcendental numbers on the real line.

The situation thus described is in marked contrast to complex differentiable functions.

[citation needed] Smooth functions with given closed support are used in the construction of smooth partitions of unity (see partition of unity and topology glossary); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence.

From what has just been said, partitions of unity do not apply to holomorphic functions; their different behavior relative to existence and analytic continuation is one of the roots of sheaf theory.

In contrast, sheaves of smooth functions tend not to carry much topological information.

Smoothness can be checked with respect to any chart of the atlas that contains

since the smoothness requirements on the transition functions between charts ensure that if

, at each point the pushforward (or differential) maps tangent vectors at

In this way smooth functions between manifolds can transport local data, like vector fields and differential forms, from one manifold to another, or down to Euclidean space where computations like integration are well understood.

Preimages and pushforwards along smooth functions are, in general, not manifolds without additional assumptions.

[10] There is a corresponding notion of smooth map for arbitrary subsets of manifolds.