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mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, the smoothness of a function is a property measured by the number of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all
orders Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
in its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function).


Differentiability classes

Differentiability class is a classification of functions according to the properties of their
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
U on the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
and a function f defined on U with real values. Let ''k'' be a non-negative
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. The function f is said to be of differentiability class ''C^k'' if the derivatives f',f'',\dots,f^ exist and are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
on U. If f is k-differentiable on U, then it is at least in the class C^ since f',f'',\dots,f^ are continuous on U. The function f is said to be infinitely differentiable, smooth, or of class C^\infty, if it has derivatives of all orders on U. (So all these derivatives are continuous functions over U.) The function f is said to be of class C^\omega, or
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
, if f is smooth (i.e., f is in the class C^\infty) and its
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion around any point in its domain converges to the function in some neighborhood of the point. C^\omega is thus strictly contained in C^\infty. Bump functions are examples of functions in C^\infty but ''not'' in C^\omega. To put it differently, the class C^0 consists of all continuous functions. The class C^1 consists of all
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
s whose derivative is continuous; such functions are called continuously differentiable. Thus, a C^1 function is exactly a function whose derivative exists and is of class C^0. In general, the classes C^k can be defined recursively by declaring C^0 to be the set of all continuous functions, and declaring C^k for any positive integer k to be the set of all differentiable functions whose derivative is in C^. In particular, C^k is contained in C^ for every k>0, and there are examples to show that this containment is strict (C^k \subsetneq C^). The class C^\infty of infinitely differentiable functions, is the intersection of the classes C^k as k varies over the non-negative integers.


Examples


Example: Continuous (''C''0) But Not Differentiable

The function f(x) = \beginx & \mbox x \geq 0, \\ 0 &\text x < 0\end is continuous, but not differentiable at , so it is of class ''C''0, but not of class ''C''1.


Example: Finitely-times Differentiable (''C'')

For each even integer , the function f(x)=, x, ^ is continuous and times differentiable at all . At , however, f is not times differentiable, so f is of class ''C'', but not of class ''C'' where .


Example: Differentiable But Not Continuously Differentiable (not ''C''1)

The function g(x) = \beginx^2\sin & \textx \neq 0, \\ 0 &\textx = 0\end is differentiable, with derivative g'(x) = \begin-\mathord + 2x\sin\left(\tfrac\right) & \textx \neq 0, \\ 0 &\textx = 0.\end Because \cos(1/x) oscillates as → 0, g'(x) is not continuous at zero. Therefore, g(x) is differentiable but not of class ''C''1.


Example: Differentiable But Not Lipschitz Continuous

The function h(x) = \beginx^\sin & \textx \neq 0, \\ 0 &\textx = 0\end is differentiable but its derivative is unbounded on a compact set. Therefore, h is an example of a function that is differentiable but not locally Lipschitz continuous.


Example: Analytic (''C'')

The
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
e^ is analytic, and hence falls into the class ''C''ω. The
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s are also analytic wherever they are defined as they are linear combinations of complex exponential functions e^ and e^.


Example: Smooth (''C'') but not Analytic (''C'')

The bump function f(x) = \begine^ & \text , x, < 1, \\ 0 &\text\end is smooth, so of class ''C'', but it is not analytic at , and hence is not of class ''C''ω. The function is an example of a smooth function with compact support.


Multivariate differentiability classes

A function f:U\subset\mathbb^n\to\mathbb defined on an open set U of \mathbb^n is said to be of class C^k on U, for a positive integer k, if all partial derivatives \frac(y_1,y_2,\ldots,y_n) exist and are continuous, for every \alpha_1,\alpha_2,\ldots,\alpha_n non-negative integers, such that \alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\leq k, and every (y_1,y_2,\ldots,y_n)\in U. Equivalently, f is of class C^k on U if the k-th order Fréchet derivative of f exists and is continuous at every point of U. The function f is said to be of class C or C^0 if it is continuous on U. Functions of class C^1 are also said to be ''continuously differentiable''. A function f:U\subset\mathbb^n\to\mathbb^m, defined on an open set U of \mathbb^n, is said to be of class C^k on U, for a positive integer k, if all of its components f_i(x_1,x_2,\ldots,x_n)=(\pi_i\circ f)(x_1,x_2,\ldots,x_n)=\pi_i(f(x_1,x_2,\ldots,x_n)) \text i=1,2,3,\ldots,m are of class C^k, where \pi_i are the natural projections \pi_i:\mathbb^m\to\mathbb defined by \pi_i(x_1,x_2,\ldots,x_m)=x_i. It is said to be of class C or C^0 if it is continuous, or equivalently, if all components f_i are continuous, on U.


The space of ''C''''k'' functions

Let D be an open subset of the real line. The set of all C^k real-valued functions defined on D is a Fréchet vector space, with the countable family of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
s p_=\sup_\left, f^(x)\ where K varies over an increasing sequence of compact sets whose union is D, and m=0,1,\dots,k. The set of C^\infty functions over D also forms a Fréchet space. One uses the same seminorms as above, except that m is allowed to range over all non-negative integer values. 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 space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s.


Continuity

The terms ''parametric continuity'' (''C''''k'') 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 In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quant ...
, with which the parameter traces out the curve.


Parametric continuity

Parametric continuity (''C''''k'') is a concept applied to parametric curves, which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve s: ,1to\mathbb^n is said to be of class ''C''''k'', if \textstyle \frac exists and is continuous on ,1/math>, where derivatives at the end-points 0,1\in ,1/math> are taken to be one sided derivatives (i.e., at 0 from the right, and at 1 from the left). As a practical application of this concept, a curve describing the motion of an object with a parameter of time must have ''C''1 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.


Order of parametric continuity

The various order of parametric continuity can be described as follows: * 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 n-th derivatives are continuous


Geometric continuity

The concept of geometrical continuity or geometric continuity (''Gn'') was primarily applied to the conic sections (and related shapes) by mathematicians such as
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
,
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
, and Poncelet. The concept was an early attempt at describing, through geometry rather than algebra, the concept of continuity as expressed through a parametric function. The basic idea behind geometric continuity was that the five conic sections were really five different versions of the same shape. An
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
tends to a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
as the eccentricity approaches zero, or to a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
as it approaches one; and a
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
tends to a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
as the eccentricity drops toward one; it can also tend to intersecting lines. Thus, there was ''continuity'' between the conic sections. These ideas led to other concepts of continuity. For instance, if a circle and a straight line were two expressions of the same shape, perhaps a line could be thought of as a circle of infinite
radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
. For such to be the case, one would have to make the line closed by allowing the point x =\infty to be a point on the circle, and for x =+\infty and x =\neg\infty to be identical. Such ideas were useful in crafting the modern, algebraically defined, idea of the continuity of a function and of \infty (see projectively extended real line for more).


Order of geometric continuity

A curve or surface can be described as having G^n continuity, with n being the increasing measure of smoothness. Consider the segments either side of a point on a curve: *G^0: The curves touch at the join point. *G^1: The curves also share a common
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
direction at the join point. *G^2: The curves also share a common center of curvature at the join point. In general, G^n continuity exists if the curves can be reparameterized to have C^n (parametric) continuity. A reparametrization of the curve is geometrically identical to the original; only the parameter is affected. Equivalently, two vector functions f(t) and g(t) have G^n continuity if f^(t)\neq0 and f^(t)\equiv kg^(t), for a scalar k>0 (i.e., if the direction, but not necessarily the magnitude, of the two vectors is equal). While it may be obvious that a curve would require G^1 continuity to appear smooth, for good
aesthetics Aesthetics, or esthetics, is a branch of philosophy that deals with the nature of beauty and taste, as well as the philosophy of art (its own area of philosophy that comes out of aesthetics). It examines aesthetic values, often expressed t ...
, such as those aspired to in
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing buildings ...
and
sports car A sports car is a car designed with an emphasis on dynamic performance, such as handling, acceleration, top speed, the thrill of driving and racing capability. Sports cars originated in Europe in the early 1900s and are currently produced by ...
design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has G^2 continuity. A (with ninety degree circular arcs at the four corners) has G^1 continuity, but does not have G^2 continuity. The same is true for a , with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with G^2 continuity is required, then cubic splines are typically chosen; these curves are frequently used in
industrial design Industrial design is a process of design applied to physical products that are to be manufactured by mass production. It is the creative act of determining and defining a product's form and features, which takes place in advance of the manufactu ...
.


Other concepts


Relation to analyticity

While all
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s 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 A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
; another example is the
Fabius function In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by . It was also written down as the Fourier transform of : \hat(z) = \prod_^\infty \left(\cos\frac\right)^m by . The ...
. 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 number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
s 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

Smooth functions with given closed
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
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 \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 , +\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 don't apply to holomorphic functions; their different behavior relative to existence and analytic continuation is one of the roots of Sheaf (mathematics), sheaf theory. In contrast, sheaves of smooth functions tend not to carry much topological information.


Smooth functions on and between manifolds

Given a smooth manifold M, of dimension m, and an
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
\mathfrak = \_\alpha, then a map f:M\to \R is smooth on M if for all p \in M there exists a chart (U, \phi) \in \mathfrak, such that p \in U, and f \circ \phi^ : \phi(U) \to \R is a smooth function from a neighborhood of \phi(p) in \R^m to \R (all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any
chart A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent ...
of the atlas that contains p, since the smoothness requirements on the transition functions between charts ensure that if fis 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^ : \phi(U) \to \psi(V) is a smooth function from \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_ : T_p M \to T_N, and on the level of the tangent bundle, the pushforward is a
vector bundle homomorphism In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ever ...
: F_* : TM \to TN. The dual to the pushforward is the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of 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 form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s, 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.


Smooth functions between subsets of manifolds

There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If f : X \to Y is a function whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
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.


See also

* * * * * * * * (number theory) * * *
Sobolev mapping In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations and partial differential equations, including the ...


References

{{Manifolds Smooth functions