Smoothness

The exponential function e−ix.

The bump function x = ±1, and hence is not of class C. The function

A function ℝn is said to be of class k, if all partial derivatives |α|=α1+α2+⋯+αn≤k. Equivalently, in finite dimensions, U if it is f is said to be of class U. Functions of class U of U, for a positive integer Ck, where πi(x1,x2,…,xm)=xi. It is said to be of class fi are continuous, on ℝn. The set of all real-valued Ck(D). With the compact-open pK,α(f)=supx∈K|Dαf(x)|, where α ranges over multi-indices with Ck(U‾) denotes the space of functions on U‾. It is a Banach space with the norm Ck(U‾) is a commutative Banach algebra. The algebra property follows from the Leibniz rule, which expresses each derivative of a product in terms of derivatives of the factors of order at most Ck(M) is a Banach space. Its norm may be defined using a finite collection of coordinate charts and a partition of unity; different such choices give equivalent norms. With pointwise multiplication, Ck(M) is Lp spaces and Sobolev spaces under suitable hypotheses. These correspond to putting topologies on the smooth functions that are weaker than those of uniform convergence (like the f and f+g, Ck. If Ck. These facts follow from the sum, product, and quotient rules for derivatives. Moreover, the space M, plays a central role in differential geometry: many geometric objects on U,V,W are open subsets of Euclidean spaces, g:V→W is of class Ck. For C∞⊆⋯⊆Ck+1⊆Ck⊆⋯⊆C1⊆C0. Thus every C1 function is continuous. On typical domains, such as open intervals or open subsets of Euclidean space, these inclusions are strict.

In several variables, continuous differentiability has several consequences for partial derivatives. If a function is of class f is of class xi and C1 is a hypothesis in local results such as the inverse function theorem and the implicit function theorem. For example, if Df(a) is invertible at a point a, and its local inverse is also of class Cc∞(U), are called test functions. They are used to define distributions and weak derivatives.

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] and such that 0 \\quad \\text{ for } \\quad a < x < b.\\,"}}'>f(x)>0 for a<x<b.

Given a locally finite collection of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals M, of dimension f:M→ℝ is smooth on (U,ϕ)∈𝔘, with ϕ(U) of f is of class p, since the smoothness requirements on the transition functions between charts ensure that if p in any other chart.

On a smooth manifold X acts on smooth functions by F:M→N is a map from N, then (U,ϕ) containing F(p) such that F:M→N, at each point the pushforward (or differential) maps tangent vectors at F∗,p:TpM→TF(p)N, and on the level of the tangent bundle, the pushforward is a vector bundle homomorphism: M, and F∗:Ωk(N)→Ωk(M). 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 images of smooth maps are, in general, not manifolds without additional assumptions. Preimages of regular values are manifolds; this means that, for a smooth map dFp:TpM→TqN is surjective at every point X⊆M and x∈X there is an open set F:U→N such that 0<α≤1, the Hölder spaces ℝn are functions that are k-th partials satisfy a Hölder condition on α=1, it implies the Lipschitz continuity of the k-th derivative, which is weaker than their differentiability. Thus, for Ck+1(U)⊊Ck,1(U)⊊Ck,α(U)⊊Ck(U).

• Discontinuity – Mathematical analysis of discontinuous pointsPages displaying short descriptions of redirect targets
• Hadamard's lemma – TheoremPages displaying short descriptions with no spaces
• Non-analytic smooth function – Mathematical functions which are smooth but not analytic
• Parametric continuity – Notion of smoothness for parametric curves
• Quasi-analytic function
• Singularity (mathematics) – Point where a mathematical object behaves irregularly
• Sinuosity – Ratio of arc length and straight-line distance between two points on a wave-like function
• Smooth scheme – Concept in algebraic geometryPages displaying short descriptions of redirect targets
• Smooth number – Integer having only small prime factors (number theory)
• Smoothing – Fitting an approximating function to data
• Spline – Mathematical function defined piecewise by polynomials
• Sobolev mapping