# Smoothness > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Smoothness > Markdown URL: https://mediated.wiki/source/Smoothness.md > Source: https://en.wikipedia.org/wiki/Smoothness > Source revision: 1356854885 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Degree of differentiability of a function or map For differentiability in general, see [differentiable function](/source/Differentiable_function). "C infinity" redirects here. For the extended complex plane C ∞ {\displaystyle \mathbb {C} _{\infty }} , see [Riemann sphere](/source/Riemann_sphere). "C^n" redirects here. For C n {\displaystyle \mathbb {C} ^{n}} , see [Complex coordinate space](/source/Complex_coordinate_space). For smoothness in number theory, see [smooth number](/source/Smooth_number). For physical smoothness, see [Surface metrology](/source/Surface_metrology), [Surface roughness](/source/Surface_roughness), and [Polishing](/source/Polishing). A [bump function](/source/Bump_function) is a smooth function with [compact support](/source/Compact_support). In [mathematical analysis](/source/Mathematical_analysis), the **smoothness** of a [function](/source/Function_(mathematics)) or [map](/source/Map_(mathematics)) describes the extent to which it has [derivatives](/source/Derivative) that exist and vary [continuously](/source/Continuous_function).[1] Given a non-negative integer k {\displaystyle k} , a function of class C k {\displaystyle C^{k}} is a function whose derivatives of all orders up to k {\displaystyle k} exist and are continuous over the function's domain. A function of class C ∞ {\displaystyle C^{\infty }} is a function that is of class C k {\displaystyle C^{k}} for every non-negative integer k {\displaystyle k} . Generally, the term **smooth function** refers to a C ∞ {\displaystyle C^{\infty }} -function. However, it may also mean "sufficiently differentiable" for the problem under consideration. For [complex-valued functions](/source/Complex-valued_function), one may still speak of C k {\displaystyle C^{k}} or C ∞ {\displaystyle C^{\infty }} smoothness by regarding the function as a map between real vector spaces. This should be distinguished from [complex differentiability](/source/Holomorphic_function): a complex function that is complex differentiable on an open subset of C {\displaystyle \mathbb {C} } is [holomorphic](/source/Holomorphic_function) and hence [analytic](/source/Analytic_function) on that set. ## Differentiability classes **Differentiability class** is a classification of functions according to the properties of their [derivatives](/source/Derivative). It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an [open set](/source/Open_set) U {\displaystyle U} on the [real line](/source/Real_line) and a function f {\displaystyle f} defined on U {\displaystyle U} with real values. Let *k* be a non-negative [integer](/source/Integer). The function f {\displaystyle f} is said to be of differentiability **class *C k {\displaystyle C^{k}}*** if the derivatives f ′ , f ″ , … , f ( k ) {\displaystyle f',f'',\dots ,f^{(k)}} exist and are [continuous](/source/Continuous_function) on U . {\displaystyle U.} If f {\displaystyle f} is of class C k {\displaystyle C^{k}} on U {\displaystyle U} and k > 0 {\displaystyle k>0} , then it is also of class C k − 1 {\displaystyle C^{k-1}} . The function f {\displaystyle f} is said to be **infinitely differentiable**, **smooth**, or of **class C ∞ , {\displaystyle C^{\infty },}** if it is of class C k {\displaystyle C^{k}} for every non-negative integer k {\displaystyle k} .[2] The function f {\displaystyle f} is said to be of **class C ω , {\displaystyle C^{\omega },}** or *[analytic](/source/Analytic_function)*, if f {\displaystyle f} is smooth and its [Taylor series](/source/Taylor_series) expansion around any point in its domain converges to the function in some [neighborhood](/source/Neighbourhood_(mathematics)) of the point. There exist functions that are smooth but not analytic; C ω {\displaystyle C^{\omega }} is thus strictly contained in C ∞ . {\displaystyle C^{\infty }.} [Bump functions](/source/Bump_function) are examples of functions with this property. To put it differently, the class C 0 {\displaystyle C^{0}} consists of all continuous functions. The class C 1 {\displaystyle C^{1}} consists of all [differentiable functions](/source/Differentiable_function) whose derivative is continuous; such functions are called *[continuously differentiable](/source/Continuously_differentiable_function)*. Thus, a C 1 {\displaystyle C^{1}} function is exactly a function whose derivative exists and is of class C 0 . {\displaystyle C^{0}.} For functions of one real variable, the classes C k {\displaystyle C^{k}} can be defined [recursively](/source/Recursion) by declaring C 0 {\displaystyle C^{0}} to be the set of all continuous functions, and declaring C k {\displaystyle C^{k}} for any positive integer k {\displaystyle k} to be the set of all differentiable functions whose derivative is in C k − 1 . {\displaystyle C^{k-1}.} In particular, C k {\displaystyle C^{k}} is contained in C k − 1 {\displaystyle C^{k-1}} for every k > 0 , {\displaystyle k>0,} and there are examples to show that this containment is strict ( C k ⊊ C k − 1 {\displaystyle C^{k}\subsetneq C^{k-1}} ). The class C ∞ {\displaystyle C^{\infty }} of infinitely differentiable functions is the intersection of the classes C k {\displaystyle C^{k}} as k {\displaystyle k} varies over the non-negative integers. ### Examples #### Continuous (*C*0) but not differentiable The *C*0 function f(x) = x for x ≥ 0 and 0 otherwise. The function g(x) = x2 sin(1/x) for x > 0. The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } with f ( x ) = x 2 sin ⁡ ( 1 x ) {\displaystyle f(x)=x^{2}\sin \left({\tfrac {1}{x}}\right)} for x ≠ 0 {\displaystyle x\neq 0} and f ( 0 ) = 0 {\displaystyle f(0)=0} is differentiable. However, this function is not continuously differentiable. A smooth function that is not analytic. The function f ( x ) = { x if x ≥ 0 , 0 if x < 0 {\displaystyle f(x)={\begin{cases}x&{\mbox{if }}x\geq 0,\\0&{\text{if }}x<0\end{cases}}} is continuous, but not differentiable at x = 0, so it is of class *C*0, but not of class *C*1. #### Finitely differentiable functions For each even non-negative integer k, the function f ( x ) = | x | k + 1 {\displaystyle f(x)=|x|^{k+1}} is continuous and of class C k {\displaystyle C^{k}} . At x = 0, however, f {\displaystyle f} is not of class C k + 1 {\displaystyle C^{k+1}} , so f {\displaystyle f} is of class *C*k, but not of class *C*j where j > k. #### Differentiable but not continuously differentiable (not *C*1) The function g ( x ) = { x 2 sin ⁡ ( 1 x ) if x ≠ 0 , 0 if x = 0 {\displaystyle g(x)={\begin{cases}x^{2}\sin {\left({\tfrac {1}{x}}\right)}&{\text{if }}x\neq 0,\\0&{\text{if }}x=0\end{cases}}} is differentiable, with derivative g ′ ( x ) = { − cos ⁡ ( 1 x ) + 2 x sin ⁡ ( 1 x ) if x ≠ 0 , 0 if x = 0. {\displaystyle g'(x)={\begin{cases}-{\mathord {\cos \left({\tfrac {1}{x}}\right)}}+2x\sin \left({\tfrac {1}{x}}\right)&{\text{if }}x\neq 0,\\0&{\text{if }}x=0.\end{cases}}} Because cos ⁡ ( 1 / x ) {\displaystyle \cos(1/x)} oscillates as x → 0, g ′ ( x ) {\displaystyle g'(x)} is not continuous at zero. Therefore, g ( x ) {\displaystyle g(x)} is differentiable but not of class *C*1. #### Differentiable but not Lipschitz continuous The function h ( x ) = { x 4 / 3 sin ⁡ ( 1 x ) if x ≠ 0 , 0 if x = 0 {\displaystyle h(x)={\begin{cases}x^{4/3}\sin {\left({\tfrac {1}{x}}\right)}&{\text{if }}x\neq 0,\\0&{\text{if }}x=0\end{cases}}} is differentiable, but its derivative is unbounded on every compact interval containing 0 {\displaystyle 0} . Therefore, h {\displaystyle h} is an example of a differentiable function that is not locally [Lipschitz continuous](/source/Lipschitz_continuous) at 0 {\displaystyle 0} . #### Analytic (*C*ω) The [exponential function](/source/Exponential_function) e x {\displaystyle e^{x}} is [analytic](/source/Analytic_function), and hence falls into the class *C*ω. The [trigonometric functions](/source/Trigonometric_function) are also analytic wherever they are defined, because they are [linear combinations of complex exponential functions](/source/Trigonometric_functions#Euler's_formula_and_the_exponential_function) e i x {\displaystyle e^{ix}} and e − i x {\displaystyle e^{-ix}} . #### Smooth (*C*∞) but not analytic (*C*ω) The [bump function](/source/Bump_function) f ( x ) = { e − 1 1 − x 2 if | x | < 1 , 0 otherwise {\displaystyle f(x)={\begin{cases}e^{-{\frac {1}{1-x^{2}}}}&{\text{ if }}|x|<1,\\0&{\text{ otherwise }}\end{cases}}} is smooth, so of class *C*∞, but it is not analytic at x = ±1, and hence is not of class *C*ω. The function f is an example of a smooth function with [compact support](/source/Compact_support). ### Multivariate differentiability classes See also: [Differentiable function § Differentiability in higher dimensions](/source/Differentiable_function#Differentiability_in_higher_dimensions) A function f : U ⊆ R n → R {\displaystyle f:U\subseteq \mathbb {R} ^{n}\to \mathbb {R} } defined on an open set U {\displaystyle U} of R n {\displaystyle \mathbb {R} ^{n}} is said[3] to be of class C k {\displaystyle C^{k}} on U {\displaystyle U} , for a positive integer k {\displaystyle k} , if all [partial derivatives](/source/Partial_derivatives) D α f = ∂ | α | f ∂ x 1 α 1 ∂ x 2 α 2 ⋯ ∂ x n α n {\displaystyle D^{\alpha }f={\frac {\partial ^{|\alpha |}f}{\partial x_{1}^{\alpha _{1}}\,\partial x_{2}^{\alpha _{2}}\,\cdots \,\partial x_{n}^{\alpha _{n}}}}} exist and are continuous for every [multi-index](/source/Multi-index) α = ( α 1 , α 2 , … , α n ) {\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})} of non-negative integers with | α | = α 1 + α 2 + ⋯ + α n ≤ k {\displaystyle |\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}\leq k} . Equivalently, in finite dimensions, f {\displaystyle f} is of class C k {\displaystyle C^{k}} on U {\displaystyle U} if it is k {\displaystyle k} times continuously [Fréchet differentiable](/source/Fr%C3%A9chet_derivative) on U {\displaystyle U} . The function f {\displaystyle f} is said to be of class C {\displaystyle C} or C 0 {\displaystyle C^{0}} if it is continuous on U {\displaystyle U} . Functions of class C 1 {\displaystyle C^{1}} are also said to be *continuously differentiable*. A function f : U ⊂ R n → R m {\displaystyle f:U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}} , defined on an open set U {\displaystyle U} of R n {\displaystyle \mathbb {R} ^{n}} , is said to be of class C k {\displaystyle C^{k}} on U {\displaystyle U} , for a positive integer k {\displaystyle k} , if all of its components f i = π i ∘ f for i = 1 , 2 , 3 , … , m {\displaystyle f_{i}=\pi _{i}\circ f\quad {\text{for }}i=1,2,3,\ldots ,m} are of class C k {\displaystyle C^{k}} , where π i {\displaystyle \pi _{i}} are the natural [projections](/source/Projection_(linear_algebra)) π i : R m → R {\displaystyle \pi _{i}:\mathbb {R} ^{m}\to \mathbb {R} } defined by π i ( x 1 , x 2 , … , x m ) = x i {\displaystyle \pi _{i}(x_{1},x_{2},\ldots ,x_{m})=x_{i}} . It is said to be of class C {\displaystyle C} or C 0 {\displaystyle C^{0}} if it is continuous, or equivalently, if all components f i {\displaystyle f_{i}} are continuous, on U {\displaystyle U} . ## Function spaces ### Open domains Let D {\displaystyle D} be an open subset of R n {\displaystyle \mathbb {R} ^{n}} . The set of all real-valued C k {\displaystyle C^{k}} functions on D {\displaystyle D} is denoted C k ( D ) {\displaystyle C^{k}(D)} . With the compact-open C k {\displaystyle C^{k}} topology, C k ( D ) {\displaystyle C^{k}(D)} is a [Fréchet space](/source/Fr%C3%A9chet_space). One way to describe this topology is by the family of [seminorms](/source/Seminorm) p K , α ( f ) = sup x ∈ K | D α f ( x ) | , {\displaystyle p_{K,\alpha }(f)=\sup _{x\in K}|D^{\alpha }f(x)|,} where K {\displaystyle K} ranges over compact subsets of D {\displaystyle D} and α {\displaystyle \alpha } ranges over multi-indices with | α | ≤ k {\displaystyle |\alpha |\leq k} . ### Compact domains If U ⊂ R n {\displaystyle U\subset \mathbb {R} ^{n}} is bounded and open, then C k ( U ¯ ) {\displaystyle C^{k}({\overline {U}})} denotes the space of functions on U {\displaystyle U} whose partial derivatives of order at most k {\displaystyle k} extend continuously to the compact set U ¯ {\displaystyle {\overline {U}}} .[4] It is a [Banach space](/source/Banach_space) with the norm ‖ f ‖ C k ( U ¯ ) = max | α | ≤ k sup x ∈ U ¯ | D α f ( x ) | . {\displaystyle \|f\|_{C^{k}({\overline {U}})}=\max _{|\alpha |\leq k}\sup _{x\in {\overline {U}}}|D^{\alpha }f(x)|.} Equivalently, one may use the sum of these suprema over | α | ≤ k {\displaystyle |\alpha |\leq k} ; the resulting norm is equivalent. Under pointwise addition and multiplication, C k ( U ¯ ) {\displaystyle C^{k}({\overline {U}})} is a commutative [Banach algebra](/source/Banach_algebra). The algebra property follows from the [Leibniz rule](/source/Leibniz_rule), which expresses each derivative of a product in terms of derivatives of the factors of order at most k {\displaystyle k} . More generally, if M {\displaystyle M} is a compact smooth manifold, possibly with boundary, then C k ( M ) {\displaystyle C^{k}(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, C k ( M ) {\displaystyle C^{k}(M)} is again a Banach algebra. By contrast, C ∞ ( M ) {\displaystyle C^{\infty }(M)} is generally not a Banach space; on a compact manifold it is naturally a [Fréchet space](/source/Fr%C3%A9chet_space), with seminorms controlling derivatives of all orders. The [Gelfand spectrum](/source/Gelfand_spectrum) of C k ( M ) {\displaystyle C^{k}(M)} is M {\displaystyle M} itself. Thus the Gelfand transform gives an injective (but not surjective) map C k ( M ) → C 0 ( M ) {\displaystyle C^{k}(M)\to C^{0}(M)} .[5]: Exercise 11.9 ### 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](/source/Partial_differential_equation), it can sometimes be more fruitful to work instead with [Sobolev spaces](/source/Sobolev_space). Smooth compactly supported functions are dense in many function spaces used in analysis, such as L p {\displaystyle L^{p}} 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 L p {\displaystyle L^{p}} norm). This makes smooth functions useful as test functions and as approximations to less regular functions. ## Basic properties The differentiability classes C k {\displaystyle C^{k}} are closed under the usual algebraic operations. If f {\displaystyle f} and g {\displaystyle g} are real-valued functions of class C k {\displaystyle C^{k}} on the same domain, then f + g {\displaystyle f+g} , f g {\displaystyle fg} , and any scalar multiple of f {\displaystyle f} are also of class C k {\displaystyle C^{k}} . If g {\displaystyle g} is nowhere zero, then the quotient f / g {\displaystyle f/g} is of class C k {\displaystyle C^{k}} . These facts follow from the sum, product, and quotient rules for derivatives.[5][6] Moreover, the space C k ( U ) {\displaystyle C^{k}(U)} is a real [vector space](/source/Vector_space) and, under pointwise multiplication, a [commutative algebra](/source/Commutative_algebra). In particular, C ∞ ( M ) {\displaystyle C^{\infty }(M)} , the algebra of smooth real-valued functions on a smooth manifold M {\displaystyle M} , plays a central role in differential geometry: many geometric objects on M {\displaystyle M} can be described in terms of their action on smooth functions. The class C k {\displaystyle C^{k}} is also closed under composition. If U , V , W {\displaystyle U,V,W} are open subsets of Euclidean spaces, f : U → V {\displaystyle f:U\to V} is of class C k {\displaystyle C^{k}} , and g : V → W {\displaystyle g:V\to W} is of class C k {\displaystyle C^{k}} , then the composite map g ∘ f : U → W {\displaystyle g\circ f:U\to W} is of class C k {\displaystyle C^{k}} . For k = 1 {\displaystyle k=1} , this is a consequence of the [chain rule](/source/Chain_rule): D ( g ∘ f ) ( x ) = D g ( f ( x ) ) ∘ D f ( x ) . {\displaystyle D(g\circ f)(x)=Dg(f(x))\circ Df(x).} The higher-order case follows by repeated differentiation.[5][6] The classes form a nested hierarchy: C ∞ ⊆ ⋯ ⊆ C k + 1 ⊆ C k ⊆ ⋯ ⊆ C 1 ⊆ C 0 . {\displaystyle C^{\infty }\subseteq \cdots \subseteq C^{k+1}\subseteq C^{k}\subseteq \cdots \subseteq C^{1}\subseteq C^{0}.} Thus every C k + 1 {\displaystyle C^{k+1}} function is C k {\displaystyle C^{k}} , and every C 1 {\displaystyle C^{1}} 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 C k {\displaystyle C^{k}} , then its mixed partial derivatives of order at most k {\displaystyle k} are independent of the order of differentiation. In particular, if f {\displaystyle f} is of class C 2 {\displaystyle C^{2}} , then ∂ 2 f ∂ x i ∂ x j = ∂ 2 f ∂ x j ∂ x i {\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}={\frac {\partial ^{2}f}{\partial x_{j}\,\partial x_{i}}}} for all coordinate directions x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} .[6] As a consequence, the [hessian matrix](/source/Hessian_matrix) of a C 2 {\displaystyle C^{2}} function is a [symmetric matrix](/source/Symmetric_matrix). The class C 1 {\displaystyle C^{1}} is a hypothesis in local results such as the [inverse function theorem](/source/Inverse_function_theorem) and the [implicit function theorem](/source/Implicit_function_theorem). For example, if f : U ⊆ R n → R n {\displaystyle f:U\subseteq \mathbb {R} ^{n}\to \mathbb {R} ^{n}} is of class C 1 {\displaystyle C^{1}} and the derivative D f ( a ) {\displaystyle Df(a)} is invertible at a point a ∈ U {\displaystyle a\in U} , then f {\displaystyle f} is locally invertible near a {\displaystyle a} , and its local inverse is also of class C 1 {\displaystyle C^{1}} .[5][6] ## Continuity The terms *parametric continuity* (*C**k*) and *geometric continuity* (*Gn*) were introduced by [Brian Barsky](/source/Brian_A._Barsky), to show that the smoothness of a curve can be measured either with respect to a particular parametrization or after allowing changes in the [speed](/source/Speed) with which the parameter traces out the curve.[7][8][9] ### Parametric continuity **Parametric continuity** (***C******k***) is a concept applied to [parametric curves](/source/Parametric_curve), which describes the smoothness of the curve as a function of its parameter. A (parametric) curve s : [ 0 , 1 ] → R n {\displaystyle s:[0,1]\to \mathbb {R} ^{n}} is said to be of class *C**k* if the derivatives of s {\displaystyle s} up to order k {\displaystyle k} exist and are continuous on [ 0 , 1 ] {\displaystyle [0,1]} , where derivatives at the end-points 0 {\displaystyle 0} and 1 {\displaystyle 1} are taken to be [one-sided derivatives](/source/Semi-differentiability) (from the right at 0 {\displaystyle 0} and from the left at 1 {\displaystyle 1} ). As a practical application of this concept, a curve describing the motion of an object with a parameter of time has *C*1 continuity when its velocity varies continuously, and *C*2 continuity when its acceleration varies continuously. For smoother motion, such as that of a camera's path while making a film, higher orders of parametric continuity may be required. #### Order of parametric continuity Two [Bézier curve](/source/B%C3%A9zier_curve) segments attached in a way that is only C0 continuous Two Bézier curve segments attached in such a way that they are C1 continuous The various orders of parametric continuity can be described as follows:[10] - C 0 {\displaystyle C^{0}} : zeroth derivative is continuous (curves are continuous) - C 1 {\displaystyle C^{1}} : zeroth and first derivatives are continuous - C 2 {\displaystyle C^{2}} : zeroth, first and second derivatives are continuous - C n {\displaystyle C^{n}} : 0-th through n {\displaystyle n} -th derivatives are continuous ### Geometric continuity "Geometric continuity" redirects here; not to be confused with [Geometrical continuity](/source/Geometrical_continuity) or [Continuous geometry](/source/Continuous_geometry). Curves with *G*1-contact (circles,line) ( 1 − ε 2 ) x 2 − 2 p x + y 2 = 0 ,   p > 0   , ε ≥ 0 {\displaystyle (1-\varepsilon ^{2})x^{2}-2px+y^{2}=0,\ p>0\ ,\varepsilon \geq 0} pencil of conic sections with *G*2-contact: p fix, ε {\displaystyle \varepsilon } variable ( ε = 0 {\displaystyle \varepsilon =0} : circle, ε = 0.8 {\displaystyle \varepsilon =0.8} : ellipse, ε = 1 {\displaystyle \varepsilon =1} : parabola, ε = 1.2 {\displaystyle \varepsilon =1.2} : hyperbola) A [curve](/source/Curve) or [surface](/source/Surface_(topology)) can be described as having G n {\displaystyle G^{n}} continuity, with n {\displaystyle n} being an increasing measure of smoothness. Consider the segments on either side of a point on a curve: - G 0 {\displaystyle G^{0}} : The curves touch at the join point. - G 1 {\displaystyle G^{1}} : The curves also share a common [tangent](/source/Tangent) direction at the join point. - G 2 {\displaystyle G^{2}} : The curves also share a common center of curvature at the join point. In general, G n {\displaystyle G^{n}} continuity holds when the curves can be reparameterized so that they have C n {\displaystyle C^{n}} parametric continuity.[11][12] A reparametrization of the curve is geometrically identical to the original; only the parameter is affected. Equivalently, two vector functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} such that f ( 1 ) = g ( 0 ) {\displaystyle f(1)=g(0)} have G n {\displaystyle G^{n}} continuity at the point where they meet if they satisfy equations known as Beta-constraints. For example, the Beta-constraints for G 4 {\displaystyle G^{4}} continuity are: - g ( 1 ) ( 0 ) = β 1 f ( 1 ) ( 1 ) g ( 2 ) ( 0 ) = β 1 2 f ( 2 ) ( 1 ) + β 2 f ( 1 ) ( 1 ) g ( 3 ) ( 0 ) = β 1 3 f ( 3 ) ( 1 ) + 3 β 1 β 2 f ( 2 ) ( 1 ) + β 3 f ( 1 ) ( 1 ) g ( 4 ) ( 0 ) = β 1 4 f ( 4 ) ( 1 ) + 6 β 1 2 β 2 f ( 3 ) ( 1 ) + ( 4 β 1 β 3 + 3 β 2 2 ) f ( 2 ) ( 1 ) + β 4 f ( 1 ) ( 1 ) {\displaystyle {\begin{aligned}g^{(1)}(0)&=\beta _{1}f^{(1)}(1)\\g^{(2)}(0)&=\beta _{1}^{2}f^{(2)}(1)+\beta _{2}f^{(1)}(1)\\g^{(3)}(0)&=\beta _{1}^{3}f^{(3)}(1)+3\beta _{1}\beta _{2}f^{(2)}(1)+\beta _{3}f^{(1)}(1)\\g^{(4)}(0)&=\beta _{1}^{4}f^{(4)}(1)+6\beta _{1}^{2}\beta _{2}f^{(3)}(1)+(4\beta _{1}\beta _{3}+3\beta _{2}^{2})f^{(2)}(1)+\beta _{4}f^{(1)}(1)\\\end{aligned}}} where β 2 {\displaystyle \beta _{2}} , β 3 {\displaystyle \beta _{3}} , and β 4 {\displaystyle \beta _{4}} are arbitrary, but β 1 {\displaystyle \beta _{1}} is constrained to be positive.[11]: 65 In the case n = 1 {\displaystyle n=1} , this reduces to f ′ ( 1 ) ≠ 0 {\displaystyle f'(1)\neq 0} and f ′ ( 1 ) = k g ′ ( 0 ) {\displaystyle f'(1)=kg'(0)} , for a scalar k > 0 {\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 G 1 {\displaystyle G^{1}} continuity to appear smooth, for good [aesthetics](/source/Aesthetics), such as those aspired to in [architecture](/source/Architecture) and [sports car](/source/Sports_car) design, higher levels of geometric continuity are required. For example, [class A surface](/source/Class_A_surface) requires G 2 {\displaystyle G^{2}} or higher continuity to ensure smooth reflections in a car body. A *[rounded rectangle](/source/Rounded_rectangle)* (with ninety-degree circular arcs at the four corners) has G 1 {\displaystyle G^{1}} continuity, but does not have G 2 {\displaystyle G^{2}} continuity. The same is true for a *[rounded cube](/source/Rounded_cube)*, with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with G 2 {\displaystyle G^{2}} continuity is required, then [cubic splines](/source/Cubic_splines) are typically chosen; these curves are frequently used in [industrial design](/source/Industrial_design). ## Other concepts ### Relation to analyticity While all [analytic functions](/source/Analytic_function) are smooth on the set on which they are analytic, examples such as [bump functions](/source/Bump_function) (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](/source/Non-analytic_smooth_function#A_smooth_function_which_is_nowhere_real_analytic) can be made by means of [Fourier series](/source/Fourier_series); another example is the [Fabius function](/source/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](/source/Meagre_set) subset of the smooth functions.[13] Furthermore, for every open subset *A* of the real line, there exist smooth functions that are analytic on *A* and nowhere else.[14] 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.[15] A [theorem](/source/Borel's_lemma) of [Émile Borel](/source/%C3%89mile_Borel) states that every [formal power series](/source/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](/source/Laplace_transform) or [Fourier transform](/source/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](/source/Paley%E2%80%93Wiener_theorem). Conversely, decay of the Fourier transform can imply differentiability or continuity properties of the original function. This is often formulated using [Sobolev spaces](/source/Sobolev_space): Fourier-transform decay gives Sobolev regularity, and the [Sobolev embedding theorem](/source/Sobolev_embedding_theorem) gives conditions under which Sobolev regularity implies classical C k {\displaystyle C^{k}} smoothness. ### Test functions and distributions Smooth compactly supported functions, usually denoted C c ∞ ( U ) {\displaystyle C_{c}^{\infty }(U)} , are called [test functions](/source/Distribution_(mathematical_analysis)#Test_functions). They are used to define [distributions](/source/Distribution_(mathematical_analysis)) and [weak derivatives](/source/Weak_derivative). ### Smooth partitions of unity Smooth functions with suitably controlled [support](/source/Support_(mathematics)), especially smooth functions with compact support, are used in the construction of **smooth partitions of unity** (see *[partition of unity](/source/Partition_of_unity)* and [topology glossary](/source/Topology_glossary)); these are essential in the study of [smooth manifolds](/source/Smooth_manifold), for example to show that [Riemannian metrics](/source/Riemannian_metric) can be defined globally starting from their local existence. A simple case is that of a *[bump function](/source/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 for a < x < b . {\displaystyle f(x)>0\quad {\text{ for }}\quad a