Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.^[1]

"C infinity" redirects here. For the extended complex plane

\mathbb {C} _{\infty }

, see Riemann sphere.

A function of class $C^{k}$ is a function of smoothness at least $k$ ; that is, a function of class $C^{k}$ is a function that has a $k$ th derivative that is continuous in its domain.

A function of class $C^{\infty }$ or $C^{\infty }$ -function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous).

Generally, the term smooth function refers to a $C^{\infty }$ -function. However, it may also mean "sufficiently differentiable" for the problem under consideration.

$C^{0}$ : zeroth derivative is continuous (curves are continuous)

$C^{1}$ : zeroth and first derivatives are continuous

$C^{2}$ : zeroth, first and second derivatives are continuous

$C^{n}$ : 0-th through $m$ -th derivatives are continuous

Other concepts[edit]

Relation to analyticity[edit]

While all analytic functions are "smooth" (i.e. have all derivatives continuous) 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, it turns out that the analytic functions are scattered very thinly among the smooth ones; more rigorously, the 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 .

It is useful to compare the situation to that of the ubiquity of transcendental numbers on the real line. Both on the real line and the set of smooth functions, the examples we come up with at first thought (algebraic/rational numbers and analytic functions) are far better behaved than the majority of cases: the transcendental numbers and nowhere analytic functions have full measure (their complements are meagre).

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

Smooth partitions of unity[edit]

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. 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 $f(x)>0\quad {\text{ for }}\quad a<x<b.\,$

Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals $(-\infty ,c]$ and $[d,+\infty )$ 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; 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.

Smooth functions on and between manifolds[edit]

Given a smooth manifold $M$ , of dimension $m,$ and an atlas ${\mathfrak {U}}=\{(U_{\alpha },\phi _{\alpha })\}_{\alpha },$ then a map $f:M\to \mathbb {R}$ is smooth on $M$ if for all $p\in M$ there exists a chart $(U,\phi )\in {\mathfrak {U}},$ such that $p\in U,$ and $f\circ \phi ^{-1}:\phi (U)\to \mathbb {R}$ is a smooth function from a neighborhood of $\phi (p)$ in $\mathbb {R} ^{m}$ to $\mathbb {R}$ (all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any chart of the atlas that contains $p,$ since the smoothness requirements on the transition functions between charts ensure that if $f$ is smooth near $p$ in one chart it will be smooth near $p$ in any other chart.

If $F:M\to N$ is a map from $M$ to an $n$ -dimensional manifold $N$ , then $F$ is smooth if, for every $p\in M,$ there is a chart $(U,\phi )$ containing $p,$ and a chart $(V,\psi )$ containing $F(p)$ such that $F(U)\subset V,$ and $\psi \circ F\circ \phi ^{-1}:\phi (U)\to \psi (V)$ is a smooth function from $\mathbb {R} ^{n}.$

Smooth maps between manifolds induce linear maps between tangent spaces: for $F:M\to N$ , at each point the pushforward (or differential) maps tangent vectors at $p$ to tangent vectors at $F(p)$ : $F_{*,p}:T_{p}M\to T_{F(p)}N,$ and on the level of the tangent bundle, the pushforward is a vector bundle homomorphism: $F_{*}:TM\to TN.$ The dual to the pushforward is the pullback, which "pulls" covectors on $N$ back to covectors on $M,$ and $k$ -forms to $k$ -forms: $F^{*}:\Omega ^{k}(N)\to \Omega ^{k}(M).$ 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 pushforwards along smooth functions are, in general, not manifolds without additional assumptions. Preimages of regular points (that is, if the differential does not vanish on the preimage) are manifolds; this is the preimage theorem. Similarly, pushforwards along embeddings are manifolds.^[10]

Smooth functions between subsets of manifolds[edit]

There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If $f:X\to Y$ is a function whose domain and range are subsets of manifolds $X\subseteq M$ and $Y\subseteq N$ respectively. $f$ is said to be smooth if for all $x\in X$ there is an open set $U\subseteq M$ with $x\in U$ and a smooth function $F:U\to N$ such that $F(p)=f(p)$ for all $p\in U\cap X.$