"C infinity" redirects here. For the extended complex plane${\displaystyle {\mathbb{C}}_{\mathrm{\infty }}}$, seeRiemann sphere.

"C^n" redirects here. For${\displaystyle {\mathbb{C}}^n}$, seeComplex coordinate space.

Inmathematical analysis, thesmoothness of afunction is a property measured by the number ofcontinuousderivatives (differentiability class) it has over itsdomain.^[1]

A function ofclass${\displaystyle C^k}$ is a function of smoothness at leastk; that is, a function of class${\displaystyle C^k}$ is a function that has akth derivative that is continuous in its domain.

A function of class${\displaystyle C^{\mathrm{\infty }}}$ or${\displaystyle C^{\mathrm{\infty }}}$-function (pronouncedC-infinity function) is aninfinitely differentiable function, that is, a function that has derivatives of allorders (this implies that all these derivatives are continuous).

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

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

Consider anopen set${\displaystyle U}$ on thereal line and a function${\displaystyle f}$ defined on${\displaystyle U}$ with real values. Letk be a non-negativeinteger. The function${\displaystyle f}$ is said to be of differentiabilityclass${\displaystyle C^k}$ if the derivatives${\displaystyle f^{\prime },f^{″},\dots ,f^{(k)}}$ exist and arecontinuous on${\displaystyle U.}$ If${\displaystyle f}$ is${\displaystyle k}$-differentiable on${\displaystyle U,}$ then it is at least in the class${\displaystyle C^{k-1}}$ since${\displaystyle f^{\prime },f^{″},\dots ,f^{(k-1)}}$ are continuous on${\displaystyle U.}$ The function${\displaystyle f}$ is said to beinfinitely differentiable,smooth, or ofclass${\displaystyle C^{\mathrm{\infty }},}$ if it has derivatives of all orders on${\displaystyle U.}$ (So all these derivatives are continuous functions over${\displaystyle U.}$)^[2] The function${\displaystyle f}$ is said to be ofclass${\displaystyle C^{\omega },}$ oranalytic, if${\displaystyle f}$ is smooth (i.e.,${\displaystyle f}$ is in the class${\displaystyle C^{\mathrm{\infty }}}$) and itsTaylor series expansion around any point in its domain converges to the function in someneighborhood of the point. There exist functions that are smooth but not analytic;${\displaystyle C^{\omega }}$ is thus strictly contained in${\displaystyle C^{\mathrm{\infty }}.}$Bump functions are examples of functions with this property.

To put it differently, the class${\displaystyle C^0}$ consists of all continuous functions. The class${\displaystyle C^1}$ consists of alldifferentiable functions whose derivative is continuous; such functions are calledcontinuously differentiable. Thus, a${\displaystyle C^1}$ function is exactly a function whose derivative exists and is of class${\displaystyle C^0.}$ In general, the classes${\displaystyle C^k}$ can be definedrecursively by declaring${\displaystyle C^0}$ to be the set of all continuous functions, and declaring${\displaystyle C^k}$ for any positive integer${\displaystyle k}$ to be the set of all differentiable functions whose derivative is in${\displaystyle C^{k-1}.}$ In particular,${\displaystyle C^k}$ is contained in${\displaystyle C^{k-1}}$ for every${\displaystyle k>0,}$ and there are examples to show that this containment is strict (${\displaystyle C^k⊊C^{k-1}}$). The class${\displaystyle C^{\mathrm{\infty }}}$ of infinitely differentiable functions, is the intersection of the classes${\displaystyle C^k}$ as${\displaystyle k}$ varies over the non-negative integers.

The functionis continuous, but not differentiable atx = 0, so it is of classC⁰, but not of classC¹.

For each even integerk, the function$${\displaystyle f(x)=|x{|}^{k+1}}$$is continuous andk times differentiable at allx. Atx = 0, however,${\displaystyle f}$ is not(k + 1) times differentiable, so${\displaystyle f}$ is of classC^k, but not of classC^j wherej >k.

The functionis differentiable, with derivative

Because${\displaystyle \cos (1/x)}$ oscillates asx → 0,${\displaystyle g^{\prime }(x)}$ is not continuous at zero. Therefore,${\displaystyle g(x)}$ is differentiable but not of classC¹.

The functionis differentiable but its derivative is unbounded on acompact set. Therefore,${\displaystyle h}$ is an example of a function that is differentiable but not locallyLipschitz continuous.

Theexponential function${\displaystyle e^x}$ isanalytic, and hence falls into the classC^ω (where ω is the smallesttransfinite ordinal). Thetrigonometric functions are also analytic wherever they are defined, because they are linear combinations of complex exponential functions${\displaystyle e^{ix}}$ and${\displaystyle e^{-ix}}$.

Thebump functionis smooth, so of classC^∞, but it is not analytic atx = ±1, and hence is not of classC^ω. The functionf is an example of a smooth function withcompact support.

A function${\displaystyle f:U\subseteq {\mathbb{R}}^n\to \mathbb{R}}$ defined on an open set${\displaystyle U}$ of${\displaystyle {\mathbb{R}}^n}$ is said^[3] to be of class${\displaystyle C^k}$ on${\displaystyle U}$, for a positive integer${\displaystyle k}$, if allpartial derivatives$${\displaystyle \frac{{\mathrm{\partial }}^{\alpha }f}{\mathrm{\partial }{x}_1^{{\alpha }_1}\mathrm{\partial }{x}_2^{{\alpha }_2}\cdots \mathrm{\partial }{x}_n^{{\alpha }_n}}(y_1,y_2,\dots ,y_n)}$$exist and are continuous, for every${\displaystyle {\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n}$ non-negative integers, such that${\displaystyle \alpha ={\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n\le k}$, and every${\displaystyle (y_1,y_2,\dots ,y_n)\in U}$. Equivalently,${\displaystyle f}$ is of class${\displaystyle C^k}$ on${\displaystyle U}$ if the${\displaystyle k}$-th orderFréchet derivative of${\displaystyle f}$ exists and is continuous at every point of${\displaystyle U}$. The function${\displaystyle f}$ is said to be of class${\displaystyle C}$ or${\displaystyle C^0}$ if it is continuous on${\displaystyle U}$. Functions of class${\displaystyle C^1}$ are also said to becontinuously differentiable.

A function${\displaystyle f:U\subset {\mathbb{R}}^n\to {\mathbb{R}}^m}$, defined on an open set${\displaystyle U}$ of${\displaystyle {\mathbb{R}}^n}$, is said to be of class${\displaystyle C^k}$ on${\displaystyle U}$, for a positive integer${\displaystyle k}$, if all of its components$${\displaystyle f_i(x_1,x_2,\dots ,x_n)=({\pi }_i\circ f)(x_1,x_2,\dots ,x_n)={\pi }_i(f(x_1,x_2,\dots ,x_n))\text{ for }i=1,2,3,\dots ,m}$$are of class${\displaystyle C^k}$, where${\displaystyle {\pi }_i}$ are the naturalprojections${\displaystyle {\pi }_i:{\mathbb{R}}^m\to \mathbb{R}}$ defined by${\displaystyle {\pi }_i(x_1,x_2,\dots ,x_m)=x_i}$. It is said to be of class${\displaystyle C}$ or${\displaystyle C^0}$ if it is continuous, or equivalently, if all components${\displaystyle f_i}$ are continuous, on${\displaystyle U}$.

Let${\displaystyle D}$ be an open subset of the real line. The set of all${\displaystyle C^k}$ real-valued functions defined on${\displaystyle D}$ is aFréchet vector space, with the countable family ofseminorms$${\displaystyle p_{K,m}=\underset{x\in K}{sup}\left|f^{(m)}(x)\right|}$$where${\displaystyle K}$ varies over an increasing sequence ofcompact sets whoseunion is${\displaystyle D}$, and${\displaystyle m=0,1,\dots ,k}$.

The set of${\displaystyle C^{\mathrm{\infty }}}$ functions over${\displaystyle D}$ also forms a Fréchet space. One uses the same seminorms as above, except that${\displaystyle m}$ 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 ofpartial differential equations, it can sometimes be more fruitful to work instead with theSobolev spaces.

The termsparametric continuity (C^k) andgeometric continuity (Gⁿ) were introduced byBrian Barsky, to show that the smoothness of a curve could be measured by removing restrictions on thespeed, with which the parameter traces out the curve.^[4][5][6]

Parametric continuity (C^k) is a concept applied toparametric curves, which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve${\displaystyle s:[0,1]\to {\mathbb{R}}^n}$ is said to be of classC^k, if${\displaystyle \frac{d^{k}s}{d{t}^k}}$ exists and is continuous on${\displaystyle [0,1]}$, where derivatives at the end-points${\displaystyle 0}$ and${\displaystyle 1}$ are taken to beone sided derivatives (from the right at${\displaystyle 0}$ and from the left at${\displaystyle 1}$).

As a practical application of this concept, a curve describing the motion of an object with a parameter of time must haveC¹ 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.

The various order of parametric continuity can be described as follows:^[7]

• ${\displaystyle C^0}$: zeroth derivative is continuous (curves are continuous)
• ${\displaystyle C^1}$: zeroth and first derivatives are continuous
• ${\displaystyle C^2}$: zeroth, first and second derivatives are continuous
• ${\displaystyle C^n}$: 0-th through${\displaystyle n}$-th derivatives are continuous

Acurve orsurface can be described as having${\displaystyle G^n}$ continuity, with${\displaystyle n}$ being the increasing measure of smoothness. Consider the segments either side of a point on a curve:

• ${\displaystyle G^0}$: The curves touch at the join point.
• ${\displaystyle G^1}$: The curves also share a commontangent direction at the join point.
• ${\displaystyle G^2}$: The curves also share a common center of curvature at the join point.

In general,${\displaystyle G^n}$ continuity exists if the curves can be reparameterized to have${\displaystyle C^n}$ (parametric) continuity.^[8][9] A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.

Equivalently, two vector functions${\displaystyle f(t)}$ and${\displaystyle g(t)}$ such that${\displaystyle f(1)=g(0)}$ have${\displaystyle G^n}$ continuity at the point where they meet ifthey satisfy equations known as Beta-constraints. For example, the Beta-constraints for${\displaystyle G^4}$ continuity are:

where${\displaystyle {\beta }_2}$,${\displaystyle {\beta }_3}$, and${\displaystyle {\beta }_4}$ are arbitrary, but${\displaystyle {\beta }_1}$ is constrained to be positive.^[8]: 65In the case${\displaystyle n=1}$, this reduces to${\displaystyle f^{\prime }(1)\ne 0}$ and${\displaystyle f^{\prime }(1)=k{g}^{\prime }(0)}$, for a scalar${\displaystyle k>0}$ (i.e., the direction, but not necessarily the magnitude, of the two vectors is equal).

While it may be obvious that a curve would require${\displaystyle G^1}$ continuity to appear smooth, for goodaesthetics, such as those aspired to inarchitecture andsports car design, higher levels of geometric continuity are required. For example,class A surface requires${\displaystyle G^2}$ or higher continuity to ensure smooth reflections in a car body.

Arounded rectangle (with ninety degree circular arcs at the four corners) has${\displaystyle G^1}$ continuity, but does not have${\displaystyle G^2}$ continuity. The same is true for arounded cube, with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with${\displaystyle G^2}$ continuity is required, thencubic splines are typically chosen; these curves are frequently used inindustrial design.

While allanalytic functions are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such asbump 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 aresmooth but not analytic at any point can be made by means ofFourier series; another example is theFabius 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 ameagre subset of the smooth functions. Furthermore, for every open subsetA of the real line, there exist smooth functions that are analytic onA and nowhere else.^{[citation needed]}

It is useful to compare the situation to that of the ubiquity oftranscendental 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.^{[citation needed]}

Smooth functions with given closedsupport are used in the construction ofsmooth partitions of unity (seepartition of unity andtopology glossary); these are essential in the study ofsmooth manifolds, for example to show thatRiemannian metrics can be defined globally starting from their local existence. A simple case is that of abump function on the real line, that is, a smooth functionf that takes the value 0 outside an interval [a,b] and such that

Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals${\displaystyle (-\mathrm{\infty },c]}$ and${\displaystyle [d,+\mathrm{\infty })}$ 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 toholomorphic functions; their different behavior relative to existence andanalytic continuation is one of the roots ofsheaf theory. In contrast, sheaves of smooth functions tend not to carry much topological information.

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

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

Smooth maps between manifolds induce linear maps betweentangent spaces: for${\displaystyle F:M\to N}$, at each point thepushforward (or differential) maps tangent vectors at${\displaystyle p}$ to tangent vectors at${\displaystyle F(p)}$:${\displaystyle F_{\ast ,p}:T_{p}M\to T_{F(p)}N,}$ and on the level of thetangent bundle, the pushforward is avector bundle homomorphism:${\displaystyle F_{\ast }:TM\to TN.}$ The dual to the pushforward is thepullback, which "pulls" covectors on${\displaystyle N}$ back to covectors on${\displaystyle M,}$ and${\displaystyle k}$-forms to${\displaystyle k}$-forms:${\displaystyle F^{\ast }:{\mathrm{\Omega }}^k(N)\to {\mathrm{\Omega }}^k(M).}$ In this way smooth functions between manifolds can transportlocal data, likevector fields anddifferential forms, from one manifold to another, or down to Euclidean space where computations likeintegration 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 thepreimage theorem. Similarly, pushforwards along embeddings are manifolds.^[10]

There is a corresponding notion ofsmooth map for arbitrary subsets of manifolds. If${\displaystyle f:X\to Y}$ is afunction whosedomain andrange are subsets of manifolds${\displaystyle X\subseteq M}$ and${\displaystyle Y\subseteq N}$ respectively.${\displaystyle f}$ is said to besmooth if for all${\displaystyle x\in X}$ there is an open set${\displaystyle U\subseteq M}$ with${\displaystyle x\in U}$ and a smooth function${\displaystyle F:U\to N}$ such that${\displaystyle F(p)=f(p)}$ for all${\displaystyle p\in U\cap X.}$

• Discontinuity – Mathematical analysis of discontinuous pointsPages displaying short descriptions of redirect targets
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• Non-analytic smooth function – Mathematical functions which are smooth but not analytic
• 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 – Scheme without singular points, generalizing smooth manifoldsPages 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

• Smooth functions

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