Editing Smoothness

{{Short description|Number of derivatives of a function (mathematics)}}
{{redirect|C infinity|the extended complex plane <math>\mathbb{C}_\infty</math>|Riemann sphere}}
{{redirect|C^n|<math>\mathbb{C}^n</math>|Complex coordinate space}}
{{for|smoothness in number theory|smooth number}}
[[Image:Bump2D illustration.png|thumb|upright=1.2|A [[bump function]] is a smooth function with [[compact support]].]]

In [[mathematical analysis]], the '''smoothness''' of a [[function (mathematics)|function]] is a property measured by the number of [[Continuous function|continuous]] [[Derivative (mathematics)|derivatives]] (''differentiability class)'' it has over its [[Domain of a function|domain]].<ref>{{Cite web|url=http://mathworld.wolfram.com/SmoothFunction.html|title=Smooth Function|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-13|archive-date=2019-12-16|archive-url=https://web.archive.org/web/20191216043114/http://mathworld.wolfram.com/SmoothFunction.html|url-status=live}}</ref> 

A function of '''class''' <math>C^k</math> is a function of smoothness at least {{mvar|k}};  that is, a function of class <math>C^k</math> is a function that has a {{mvar|k}}th derivative that is continuous in its domain. 

A function of class <math>C^\infty</math> or <math>C^\infty</math>-function (pronounced '''C-infinity function''') is an '''infinitely   differentiable function''', that is, a function that has derivatives of all [[Order of derivation|orders]] (this implies that all these derivatives are continuous).   

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

==Differentiability classes==
'''Differentiability class''' is a classification of functions according to the properties of their [[derivative]]s. It is a measure of the highest order of derivative that exists and is continuous for a function.

Consider an [[open set]] <math>U</math> on the [[real line]] and a function <math>f</math> defined on <math>U</math> with real values. Let ''k'' be a non-negative [[integer]]. The function <math>f</math> is said to be of differentiability '''class ''<math>C^k</math>''''' if the derivatives <math>f',f'',\dots,f^{(k)}</math> exist and are [[continuous function|continuous]] on <math>U.</math> If <math>f</math> is <math>k</math>-differentiable on <math>U,</math> then it is at least in the class <math>C^{k-1}</math> since <math>f',f'',\dots,f^{(k-1)}</math> are continuous on <math>U.</math> The function <math>f</math> is said to be '''infinitely differentiable''', '''smooth''', or of '''class <math>C^\infty,</math>''' if it has derivatives of all orders on <math>U.</math> (So all these derivatives are continuous functions over <math>U.</math>)<ref name="def diff">{{cite book| last=Warner| first=Frank W.| author-link=Frank Wilson Warner| year=1983| title=Foundations of Differentiable Manifolds and Lie Groups| publisher=Springer| isbn=978-0-387-90894-6| page=5 [Definition 1.2]| url=https://books.google.com/books?id=t6PNrjnfhuIC&dq=%22f+is+differentiable+of+class+Ck%22&pg=PA5| access-date=2014-11-28| archive-date=2015-10-01| archive-url=https://web.archive.org/web/20151001012659/https://books.google.com/books?id=t6PNrjnfhuIC&pg=PA5&dq=%22f+is+differentiable+of+class+Ck%22| url-status=live}}</ref> The function <math>f</math> is said to be of '''class <math>C^\omega,</math>''' or ''[[analytic function|analytic]]'', if <math>f</math> is smooth (i.e., <math>f</math> is in the class <math>C^\infty</math>) and its [[Taylor series]] expansion around any point in its domain converges to the function in some [[Neighbourhood (mathematics)|neighborhood]] of the point.  There exist functions that are smooth but not analytic; <math>C^\omega</math> is thus strictly contained in <math>C^\infty.</math> [[Bump function]]s are examples of functions with this property.

To put it differently, the class <math>C^0</math> consists of all continuous functions. The class <math>C^1</math> consists of all [[differentiable function]]s whose derivative is continuous; such functions are called ''continuously differentiable''. Thus, a <math>C^1</math> function is exactly a function whose derivative exists and is of class <math>C^0.</math> In general, the classes <math>C^k</math> can be defined [[recursion|recursively]] by declaring <math>C^0</math> to be the set of all continuous functions, and declaring <math>C^k</math> for any positive integer <math>k</math> to be the set of all differentiable functions whose derivative is in <math>C^{k-1}.</math> In particular, <math>C^k</math> is contained in <math>C^{k-1}</math> for every <math>k>0,</math> and there are examples to show that this containment is strict (<math>C^k \subsetneq C^{k-1}</math>). The class <math>C^\infty</math> of infinitely differentiable functions, is the intersection of the classes <math>C^k</math> as <math>k</math> varies over the non-negative integers.

===Examples===
==== Example: continuous (''C''<sup>0</sup>) but not differentiable ====
[[Image:C0 function.svg|thumb|The ''C''<sup>0</sup> function {{nowrap|1={{mvar|f}}({{mvar|x}}) = {{mvar|x}}}} for {{nowrap|{{mvar|x}} ≥ 0}} and 0 otherwise.]]
[[File:X^2sin(x^-1).svg|thumb|The function {{nowrap|1={{mvar|g}}({{mvar|x}}) = {{mvar|x}}<sup>2</sup> sin(1/{{mvar|x}})}} for {{nowrap|{{mvar|x}} &gt; 0}}.]]
[[File:The function x^2*sin(1 over x).svg|thumb|upright=1.3|The function <math>f:\R\to\R</math> with <math>f(x)=x^2\sin\left(\tfrac 1x\right)</math> for <math>x\neq 0</math> and <math>f(0)=0</math> is differentiable. However, this function is not continuously differentiable.]]
[[File:Mollifier Illustration.svg|thumb|upright=1.2|A smooth function that is not analytic.]]
The function
<math display="block">f(x) = \begin{cases}x  & \mbox{if } x \geq 0, \\ 0 &\text{if } x < 0\end{cases}</math>
is continuous, but not differentiable at {{nowrap|1={{mvar|x}} = 0}}, so it is of class ''C''<sup>0</sup>, but not of class ''C''<sup>1</sup>.

==== Example: finitely-times differentiable (''C''<sup>{{mvar|k}}</sup>) ====
For each even integer {{mvar|k}}, the function
<math display="block">f(x)=|x|^{k+1}</math>
is continuous and {{mvar|k}} times differentiable at all {{mvar|x}}. At {{nowrap|1={{mvar|x}} = 0}}, however, <math>f</math> is not {{nowrap|({{mvar|k}} + 1)}} times differentiable, so <math>f</math> is of class ''C''<sup>{{mvar|k}}</sup>, but not of class ''C''<sup>{{mvar|j}}</sup> where {{nowrap|{{mvar|j}} > {{mvar|k}}}}.

==== Example: differentiable but not continuously differentiable (not ''C''<sup>1</sup>)====
The function
<math display="block">g(x) = \begin{cases}x^2\sin{\left(\tfrac{1}{x}\right)} & \text{if }x \neq 0, \\ 0 &\text{if }x = 0\end{cases}</math>
is differentiable, with derivative
<math display="block">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}</math>

Because <math>\cos(1/x)</math> oscillates as {{mvar|x}} → 0, <math>g'(x)</math> is not continuous at zero. Therefore, <math>g(x)</math> is differentiable but not of class ''C''<sup>1</sup>.

==== Example: differentiable but not Lipschitz continuous ====
The function
<math display="block">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}</math>
is differentiable but its derivative is unbounded on a [[compact set]]. Therefore, <math>h</math> is an example of a function  that is differentiable but not locally [[Lipschitz continuous]].

==== Example: analytic (''C''<sup>{{mvar|ω}}</sup>) ====
The [[exponential function]] <math>e^{x}</math> is [[Analytic function|analytic]], and hence falls into the class ''C''<sup>ω</sup> (where ω is the smallest [[transfinite ordinal]]). The [[trigonometric function]]s are also analytic wherever they are defined, because they are [[Trigonometric_functions#Euler's_formula_and_the_exponential_function | linear combinations of complex exponential functions]] <math>e^{ix}</math> and <math>e^{-ix}</math>.

==== Example: smooth (''C''<sup>{{mvar|∞}}</sup>) but not analytic (''C''<sup>{{mvar|ω}}</sup>) ====
The [[bump function]]
<math display="block">f(x) = \begin{cases}e^{-\frac{1}{1-x^2}} & \text{ if } |x| < 1, \\ 0 &\text{ otherwise }\end{cases}</math>
is smooth, so of class ''C''<sup>∞</sup>, but it is not analytic at {{nowrap|1={{mvar|x}} = ±1}}, and hence is not of class ''C''<sup>ω</sup>. The function {{mvar|f}} is an example of a smooth function with [[compact support]].

===Multivariate differentiability classes===
A function <math>f:U\subseteq\mathbb{R}^n\to\mathbb{R}</math> defined on an open set <math>U</math> of <math>\mathbb{R}^n</math> is said<ref>{{cite book|author=Henri Cartan|title=Cours de calcul différentiel|year=1977|publisher=Paris: Hermann|author-link=Henri Cartan}}</ref> to be of class <math>C^k</math> on <math>U</math>, for a positive integer <math>k</math>, if all [[partial derivatives]] 
<math display="block">\frac{\partial^\alpha f}{\partial x_1^{\alpha_1} \, \partial x_2^{\alpha_2}\,\cdots\,\partial x_n^{\alpha_n}}(y_1,y_2,\ldots,y_n)</math>
exist and are continuous, for every <math>\alpha_1,\alpha_2,\ldots,\alpha_n</math> non-negative integers, such that <math>\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\leq k</math>, and every <math>(y_1,y_2,\ldots,y_n)\in U</math>. Equivalently, <math>f</math> is of class <math>C^k</math> on <math>U</math> if the <math>k</math>-th order [[Fréchet derivative]] of <math>f</math> exists and is continuous at every point of <math>U</math>. The function <math>f</math> is said to be of class <math>C</math> or <math>C^0</math> if it is continuous on <math>U</math>. Functions of class <math>C^1</math> are also said to be ''continuously differentiable''.

A function <math>f:U\subset\mathbb{R}^n\to\mathbb{R}^m</math>, defined on an open set <math>U</math> of <math>\mathbb{R}^n</math>, is said to be of class <math>C^k</math> on <math>U</math>, for a positive integer <math>k</math>, if all of its components
<math display="block">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{ for } i=1,2,3,\ldots,m</math>
are of class <math>C^k</math>, where <math>\pi_i</math> are the natural [[Projection (linear algebra)|projections]] <math>\pi_i:\mathbb{R}^m\to\mathbb{R}</math> defined by <math>\pi_i(x_1,x_2,\ldots,x_m)=x_i</math>. It is said to be of class <math>C</math> or <math>C^0</math> if it is continuous, or equivalently, if all components <math>f_i</math> are continuous, on <math>U</math>.

===The space of ''C''<sup>''k''</sup> functions===
Let <math>D</math> be an open subset of the real line. The set of all <math>C^k</math> real-valued functions defined on <math>D</math> is a [[Fréchet space|Fréchet vector space]], with the countable family of [[seminorm]]s
<math display="block">p_{K, m}=\sup_{x\in K}\left|f^{(m)}(x)\right|</math>
where <math>K</math> varies over an increasing sequence of [[compact set]]s whose [[union (set theory)|union]] is <math>D</math>, and <math>m=0,1,\dots,k</math>.

The set of <math>C^\infty</math> functions over <math>D</math> also forms a Fréchet space. One uses the same seminorms as above, except that <math>m</math> 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 equation]]s, it can sometimes be more fruitful to work instead with the [[Sobolev space]]s.

==Continuity==

The terms ''parametric continuity'' (''C''<sup>''k''</sup>) and ''geometric continuity'' (''G<sup>n</sup>'') were introduced by [[Brian A. Barsky|Brian Barsky]], to show that the smoothness of a curve could be measured by removing restrictions on the [[speed]], with which the parameter traces out the curve.<ref name="Barsky1981">{{cite thesis |type=Ph.D. |last=Barsky |first=Brian A. |date=1981 |title=The Beta-spline: A Local Representation Based on Shape Parameters and Fundamental Geometric Measures |publisher=University of Utah, Salt Lake City, Utah|url=https://dl.acm.org/citation.cfm?id=910231 }}</ref><ref name="Barsky1988">{{cite book|author=Brian A. Barsky|title=Computer Graphics and Geometric Modeling Using Beta-splines|year=1988|publisher=Springer-Verlag, Heidelberg|isbn=978-3-642-72294-3}}</ref><ref name="BartelsBeattyBarsky1987">{{cite book|author1=Richard H. Bartels|author2=John C. Beatty|author3=Brian A. Barsky|title=An Introduction to Splines for Use in Computer Graphics and Geometric Modeling|year=1987|publisher=Morgan Kaufmann|isbn=978-1-55860-400-1|at=Chapter 13. Parametric vs. Geometric Continuity}}</ref>

===Parametric continuity===
'''Parametric continuity''' ('''''C'''''<sup>'''''k'''''</sup>) is a concept applied to [[parametric curve]]s, which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve <math>s:[0,1]\to\mathbb{R}^n</math> is said to be of class ''C''<sup>''k''</sup>, if <math>\textstyle \frac{d^ks}{dt^k}</math> exists and is continuous on <math>[0,1]</math>, where derivatives at the end-points <math>0</math> and <math>1</math> are taken to be [[Semi-differentiability|one sided derivatives]] (from the right at <math>0</math> and from the left at <math>1</math>).

As a practical application of this concept, a curve describing the motion of an object with a parameter of time must have ''C''<sup>1</sup> 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====
[[File:Parametric continuity C0.svg|upright=1.2|thumb|Two [[Bézier curve]] segments attached that is only C<sup>0</sup> continuous]]
[[File:Parametric continuity vector.svg|upright=1.2|thumb|Two Bézier curve segments attached in such a way that they are C<sup>1</sup> continuous]]
The various order of parametric continuity can be described as follows:<ref>{{cite web |first=Michiel |last=van de Panne |url=https://www.cs.helsinki.fi/group/goa/mallinnus/curves/curves.html |title=Parametric Curves |work=Fall 1996 Online Notes |date=1996 |publisher=University of Toronto, Canada |access-date=2019-09-01 |archive-date=2020-11-26 |archive-url=https://web.archive.org/web/20201126212511/https://www.cs.helsinki.fi/group/goa/mallinnus/curves/curves.html |url-status=live }}</ref>
* <math>C^0</math>: zeroth derivative is continuous (curves are continuous)
* <math>C^1</math>: zeroth and first derivatives are continuous
* <math>C^2</math>: zeroth, first and second derivatives are continuous
* <math>C^n</math>: 0-th through <math>n</math>-th derivatives are continuous

===Geometric continuity===

{{Distinguish|Geometrical continuity}}
[[File:Curves g1 contact.svg|upright=1.2|thumb|Curves with ''G''<sup>1</sup>-contact (circles,line)]]
[[File:Kegelschnitt-Schar.svg|upright=1.2|thumb|<math>(1-\varepsilon^2) x^2 -2px+y^2=0 , \ p>0 \ , \varepsilon\ge 0</math><br />
pencil of conic sections with ''G''<sup>2</sup>-contact: p fix, <math>\varepsilon</math> variable <br />
(<math>\varepsilon=0</math>: circle,<math>\varepsilon=0.8</math>: ellipse, <math>\varepsilon=1</math>: parabola, <math>\varepsilon=1.2</math>: hyperbola)]]

A [[curve]] or [[Surface (topology)|surface]] can be described as having <math>G^n</math> continuity, with <math>n</math> being the increasing measure of smoothness. Consider the segments either side of a point on a curve:

*<math>G^0</math>: The curves touch at the join point.
*<math>G^1</math>: The curves also share a common [[tangent]] direction at the join point.
*<math>G^2</math>: The curves also share a common center of curvature at the join point.

In general, <math>G^n</math> continuity exists if the curves can be reparameterized to have <math>C^n</math> (parametric) continuity.<ref name=Barsky-DeRose>{{cite journal |first1=Brian A. |last1=Barsky |first2=Tony D. |last2=DeRose |title=Geometric Continuity of Parametric Curves: Three Equivalent Characterizations |journal=IEEE Computer Graphics and Applications |volume=9 |issue=6 |year=1989 |pages=60–68 |doi=10.1109/38.41470 |s2cid=17893586 }}</ref><ref>{{cite web |url=https://www2.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf#page=55 |first=Erich |last=Hartmann |title=Geometry and Algorithms for Computer Aided Design |page=55 |date=2003 |publisher=[[Technische Universität Darmstadt]] |access-date=2019-08-31 |archive-date=2020-10-23 |archive-url=https://web.archive.org/web/20201023054532/http://www2.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf#page=55 |url-status=live }}</ref> A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.

Equivalently, two vector functions <math>f(t)</math> and <math>g(t)</math> such that <math>f(1)=g(0)</math> have <math>G^n</math> continuity at the point where they meet if
they satisfy equations known as Beta-constraints.  For example, the Beta-constraints for <math>G^4</math> continuity are:

:<math>
\begin{align}
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{align}
</math>
where <math>\beta_2</math>, <math>\beta_3</math>, and <math>\beta_4</math> are arbitrary, but <math>\beta_1</math> is constrained to be positive.{{r|Barsky-DeRose|p=65}}
In the case <math>n=1</math>, this reduces to 
<math>f'(1)\neq0</math> and <math>f'(1) =  kg'(0)</math>, for a scalar <math>k>0</math> (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 <math>G^1</math> continuity to appear smooth, for good [[aesthetics]], such as those aspired to in [[architecture]] and [[sports car]] design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has <math>G^2</math> continuity.{{cn|date=April 2024}}

A {{em|[[rounded rectangle]]}} (with ninety degree circular arcs at the four corners) has <math>G^1</math> continuity, but does not have <math>G^2</math> continuity. The same is true for a {{em|[[rounded cube]]}}, with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with <math>G^2</math> continuity is required, then [[cubic splines]] are typically chosen; these curves are frequently used in [[industrial design]].

==Other concepts==

===Relation to analyticity===
While all [[analytic function]]s are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such as [[bump function]]s (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 [[Non-analytic smooth function#A smooth function which is nowhere real analytic|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 set|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.{{citation needed|date=December 2020}}

It is useful to compare the situation to that of the ubiquity of [[transcendental number]]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.{{citation needed|date=December 2020}}

===Smooth partitions of unity===
Smooth functions with given closed [[Support (mathematics)|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 manifold]]s, for example to show that [[Riemannian metric]]s 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
<math display="block">f(x) > 0 \quad \text{ for } \quad a < x < b.\,</math>

Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals <math>(-\infty, c]</math> and <math>[d, +\infty)</math> 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 function]]s; 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 [[Differentiable manifold|smooth manifold]] <math>M</math>, of dimension <math>m,</math> and an [[Atlas (topology)|atlas]] <math>\mathfrak{U} = \{(U_\alpha,\phi_\alpha)\}_\alpha,</math> then a map <math>f:M\to \R</math> is '''smooth''' on <math>M</math> if for all <math>p \in M</math> there exists a chart <math>(U, \phi) \in \mathfrak{U},</math> such that <math>p \in U,</math>  and <math>f \circ \phi^{-1} : \phi(U) \to \R</math> is a smooth function from a neighborhood of <math>\phi(p)</math> in <math>\R^m</math> to <math>\R</math> (all partial derivatives up to a given order are continuous).  Smoothness can be checked with respect to any [[Chart (topology)|chart]] of the atlas that contains <math>p,</math> since the smoothness requirements on the transition functions between charts ensure that if <math>f</math> is smooth near <math>p</math> in one chart it will be smooth near <math>p</math> in any other chart.

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

Smooth maps between manifolds induce linear maps between [[tangent space]]s: for <math>F : M \to N</math>, at each point the [[Pushforward (differential)|pushforward]] (or differential) maps tangent vectors at <math>p</math> to tangent vectors at <math>F(p)</math>: <math>F_{*,p} : T_p M \to T_{F(p)}N,</math> and on the level of the [[tangent bundle]], the pushforward is a [[vector bundle homomorphism]]: <math>F_* : TM \to TN.</math> The dual to the pushforward is the [[Pullback (differential geometry)|pullback]], which "pulls" covectors on <math>N</math> back to covectors on <math>M,</math> and <math>k</math>-forms to <math>k</math>-forms: <math>F^* : \Omega^k(N) \to \Omega^k(M).</math> In this way smooth functions between manifolds can transport [[Sheaf (mathematics)|local data]], like [[vector field]]s and [[differential form]]s, from one manifold to another, or down to Euclidean space where computations like [[Integration on manifolds|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.<ref>{{cite book |last1=Guillemin |first1=Victor |last2=Pollack |first2=Alan |title=Differential Topology |location=Englewood Cliffs |publisher=Prentice-Hall |year=1974 |isbn=0-13-212605-2 }}</ref>

===Smooth functions between subsets of manifolds===

There is a corresponding notion of '''smooth map''' for arbitrary subsets of manifolds.  If <math>f : X \to Y</math> is a [[Function (mathematics)|function]] whose [[Domain of a function|domain]] and [[Range of a function|range]] are subsets of manifolds <math>X \subseteq M</math> and <math>Y \subseteq N</math> respectively. <math>f</math> is said to be '''smooth''' if for all <math>x \in X</math> there is an open set <math>U \subseteq M</math> with <math>x \in U</math> and a smooth function <math>F : U \to N</math> such that <math>F(p) = f(p)</math> for all <math>p \in U \cap X.</math>

==See also==
* {{annotated link|Discontinuity (mathematics)|Discontinuity}}
* {{annotated link|Hadamard's lemma}}
* {{annotated link|Non-analytic smooth function}}
* {{annotated link|Quasi-analytic function}}
* {{annotated link|Singularity (mathematics)}}
* {{annotated link|Sinuosity}}
* {{annotated link|Smooth point|Smooth scheme}}
* {{annotated link|Smooth number}} (number theory)
* {{annotated link|Smoothing}}
* {{annotated link|Spline (mathematics)|Spline}}
* [[Sobolev mapping]]

==References==

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[[Category:Smooth functions]]