In mathematical analysis, the smoothness describes the number of times a function can be differentiated without producing discontinuities. The smoothness, or differentiability. For example, the absolute value, because it is continuous, but not differentiable. Generally, the term smooth function-function, that is a function having derivatives of all orders. However, it may also mean "sufficiently differentiable" for the problem under consideration.

The usual definition is local and is therefore first made for functions defined on open subsets of Euclidean space. For functions on closed intervals, closures of open sets, or more general subsets, the same notation is also used, but its meaning depends on an additional convention, such as requiring derivatives to extend continuously to the boundary or requiring the function to be locally the restriction of a smooth function defined on an open neighborhood.

Differentiability classes are used in mathematical analysis to describe different degrees of regularity for partial differential equations. They are used in differential topology to define different classes of differentiable manifolds. For complex-valued functions smoothness by regarding the function as a map between real vector spaces. This should be distinguished from complex differentiability is holomorphic and hence analytic on that set.

Differentiability classes

Differentiability class is a classification of functions according to the highest order of derivative that exists and is continuous for a function.

Consider an open set on the real line with real values. Let k be a non-negative integer is said to be of differentiability class exist and are continuous is said to be infinitely differentiable, smooth, or of . is said to be of or analytic is smooth and its Taylor series expansion around any point in its domain converges to the function in some neighborhood Bump functions are examples of functions with this property.

consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable can be defined recursively varies over the non-negative integers.

Examples

Continuous (C0) but not differentiable

is continuous, but not differentiable at , so it is of class C0, but not of class C1.

Finitely differentiable functions

For each even non-negative integer . At is of class C, but not of class C where .

Differentiable but not continuously differentiable (not C1)

oscillates as is differentiable but not of class C1.

Differentiable but not Lipschitz continuous

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

Analytic (Cω)

The exponential function is analytic, and hence falls into the class Cω. The trigonometric functions are also analytic wherever they are defined, because they are linear combinations of complex exponential functions.

Smooth (C∞) but not analytic (Cω)

The bump function is smooth, so of class C∞, but it is not analytic at , and hence is not of class Cω. The function is an example of a smooth function with compact support.

Multivariate differentiability classes

is said, if all partial derivatives exist and are continuous for every multi-index times continuously Fréchet differentiable are also said to be continuously differentiable.

are the natural projections.

Function spaces

Open domains

is a Fréchet space. One way to describe this topology is by the family of seminorms.

Compact domains

. It is a Banach space; the resulting norm is equivalent.

is a commutative Banach algebra. The algebra property follows from the Leibniz rule.

is generally not a Banach space; on a compact manifold it is naturally a Fréchet space, with seminorms controlling derivatives of all orders.

The Gelfand spectrum.

Density

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 Sobolev spaces.

norm). This makes smooth functions useful as test functions and as approximations to less regular functions.

Basic properties

. These facts follow from the sum, product, and quotient rules for derivatives. is a real vector space and, under pointwise multiplication, a commutative algebra can be described in terms of their action on smooth functions.

, this is a consequence of the chain rule The higher-order case follows by repeated differentiation.

function is continuous. On typical domains, such as open intervals or open subsets of Euclidean space, these inclusions are strict.

. As a consequence, the hessian matrix function is a symmetric matrix.

is a hypothesis in local results such as the inverse function theorem and the implicit function theorem.

Other concepts

Relation to analyticity

While all analytic functions are smooth on the set on which they are analytic, examples such as bump functions (mentioned above) show that the converse is not true for functions on the reals: there exist smooth real functions that are not analytic. 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. Although it might seem that such functions are the exception rather than the rule, analytic functions form a small subclass of smooth functions; for example, with suitable topologies on spaces of smooth functions, analytic functions form a meagre subset of the smooth functions. Furthermore, for every open subset A of the real line, there exist smooth functions that are analytic on A and nowhere else.

The situation thus described is in marked contrast to complex differentiable functions. If a complex function is holomorphic on an open set, it is infinitely differentiable and analytic on that set.

A theorem of Émile Borel states that every formal power series occurs as the Taylor series of some smooth function. This is another way in which smooth functions differ from analytic functions, whose Taylor series determine them locally.

Smoothness and the Fourier transform

Under suitable hypotheses, higher differentiability of a function is related to faster decay of its Laplace transform or Fourier transform. For example, integration by parts gives decay estimates for Fourier transforms of functions whose derivatives satisfy appropriate integrability or boundary conditions. These relationships are related to results such as the Paley–Wiener theorem.

Conversely, decay of the Fourier transform can imply differentiability or continuity properties of the original function. This is often formulated using Sobolev spaces: Fourier-transform decay gives Sobolev regularity, and the Sobolev embedding theorem smoothness.

Test functions and distributions

, are called test functions. They are used to define distributions and weak derivatives.

Smooth partitions of unity

Smooth functions with suitably controlled support, especially smooth functions with compact 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. A simple case is that of a bump function on the real line, that is, a smooth function f that takes the value 0 outside an interval [a,b

to cover the whole line, such that the sum of the functions is always 1.

From what has just been said, partitions of unity do not apply to holomorphic functions in the same way; for example, there are no nonzero holomorphic functions with compact support on a connected complex domain. Their different behavior relative to existence and analytic continuation is one of the roots of sheaf theory. In contrast, sheaves of smooth functions are fine and hence have different cohomological behavior.

Smooth functions on and between manifolds

Given a smooth manifold and an atlas is smooth. Smoothness can be checked with respect to any chart in any other chart.

, smooth vector fields can be identified with derivations

is a smooth function between open subsets of Euclidean spaces.

Smooth maps between manifolds induce linear maps between tangent spaces, at each point the pushforward and on the level of the tangent bundle, the pushforward is a vector bundle homomorphism The dual to the pushforward is the pullback 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.

. This is the preimage theorem. Similarly, the image of an embedding is an embedded submanifold.

Smoothness is also defined for sections of vector bundles. A section is smooth if its coordinate components are smooth in local trivializations. Smooth vector fields, differential forms, and tensor fields are examples of smooth sections.

Smooth functions between subsets of manifolds

There is a corresponding notion of smooth map is a function whose domain and codomain is said to be smooth

Hölder spaces

, the Hölder spaces-th partials satisfy a Hölder condition, it implies the Lipschitz continuity

See also

• (number theory)

• Sobolev mapping

References