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

A bump function is a smooth function with compact support.

A function of class is a function of smoothness at least k; that is, a function of class is a function that has a kth derivative that is continuous in its domain.

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

Generally, the term smooth function refers to a -function. However, it may also mean "sufficiently differentiable" for the problem under consideration.

Differentiability classes

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

Consider an open set   on the real line and a function   defined on   with real values. Let k be a non-negative integer. The function   is said to be of differentiability class   if the derivatives   exist and are continuous on   If   is  -differentiable on   then it is at least in the class   since   are continuous on   The function   is said to be infinitely differentiable, smooth, or of class   if it has derivatives of all orders on   (So all these derivatives are continuous functions over  )[2] The function   is said to be of class   or analytic, if   is smooth (i.e.,   is in the class  ) and its Taylor series expansion around any point in its domain converges to the function in some neighborhood of the point. There exist functions that are smooth but not analytic;   is thus strictly contained in   Bump functions are examples of functions with this property.

To put it differently, the class   consists of all continuous functions. The class   consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a   function is exactly a function whose derivative exists and is of class   In general, the classes   can be defined recursively by declaring   to be the set of all continuous functions, and declaring   for any positive integer   to be the set of all differentiable functions whose derivative is in   In particular,   is contained in   for every   and there are examples to show that this containment is strict ( ). The class   of infinitely differentiable functions, is the intersection of the classes   as   varies over the non-negative integers.

Examples

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Example: Continuous (C0) But Not Differentiable

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The C0 function f(x) = x for x ≥ 0 and 0 otherwise.
 
The function g(x) = x2 sin(1/x) for x > 0.
 
The function   with   for   and   is differentiable. However, this function is not continuously differentiable.
 
A smooth function that is not analytic.

The function

 
is continuous, but not differentiable at x = 0, so it is of class C0, but not of class C1.

Example: Finitely-times Differentiable (Ck)

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For each even integer k, the function

 
is continuous and k times differentiable at all x. At x = 0, however,   is not (k + 1) times differentiable, so   is of class Ck, but not of class Cj where j > k.

Example: Differentiable But Not Continuously Differentiable (not C1)

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The function

 
is differentiable, with derivative
 

Because   oscillates as x → 0,   is not continuous at zero. Therefore,   is differentiable but not of class C1.

Example: Differentiable But Not Lipschitz Continuous

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The function

 
is differentiable but its derivative is unbounded on a compact set. Therefore,   is an example of a function that is differentiable but not locally Lipschitz continuous.

Example: Analytic (Cω)

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The exponential function   is analytic, and hence falls into the class Cω. The trigonometric functions are also analytic wherever they are defined, because they are linear combinations of complex exponential functions   and  .

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

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The bump function

 
is smooth, so of class C, but it is not analytic at x = ±1, and hence is not of class Cω. The function f is an example of a smooth function with compact support.

Multivariate differentiability classes

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A function   defined on an open set   of   is said[3] to be of class   on  , for a positive integer  , if all partial derivatives

 
exist and are continuous, for every   non-negative integers, such that  , and every  . Equivalently,   is of class   on   if the  -th order Fréchet derivative of   exists and is continuous at every point of  . The function   is said to be of class   or   if it is continuous on  . Functions of class   are also said to be continuously differentiable.

A function  , defined on an open set   of  , is said to be of class   on  , for a positive integer  , if all of its components

 
are of class  , where   are the natural projections   defined by  . It is said to be of class   or   if it is continuous, or equivalently, if all components   are continuous, on  .

The space of Ck functions

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Let   be an open subset of the real line. The set of all   real-valued functions defined on   is a Fréchet vector space, with the countable family of seminorms

 
where   varies over an increasing sequence of compact sets whose union is  , and  .

The set of   functions over   also forms a Fréchet space. One uses the same seminorms as above, except that   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 spaces.

Continuity

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The terms parametric continuity (Ck) 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, with which the parameter traces out the curve.[4][5][6]

Parametric continuity

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Parametric continuity (Ck) is a concept applied to parametric curves, which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve   is said to be of class Ck, if   exists and is continuous on  , where derivatives at the end-points   and   are taken to be one sided derivatives (from the right at   and from the left at  ).

As a practical application of this concept, a curve describing the motion of an object with a parameter of time must have C1 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

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Two Bézier curve segments attached that is only C0 continuous
 
Two Bézier curve segments attached in such a way that they are C1 continuous

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

  •  : zeroth derivative is continuous (curves are continuous)
  •  : zeroth and first derivatives are continuous
  •  : zeroth, first and second derivatives are continuous
  •  : 0-th through  -th derivatives are continuous

Geometric continuity

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Curves with G1-contact (circles,line)
 
 
pencil of conic sections with G2-contact: p fix,   variable
( : circle, : ellipse,  : parabola,  : hyperbola)

A curve or surface can be described as having   continuity, with   being the increasing measure of smoothness. Consider the segments either side of a point on a curve:

  •  : The curves touch at the join point.
  •  : The curves also share a common tangent direction at the join point.
  •  : The curves also share a common center of curvature at the join point.

In general,   continuity exists if the curves can be reparameterized to have   (parametric) continuity.[8][9] A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.

Equivalently, two vector functions   and   such that   have   continuity at the point where they meet if they satisfy equations known as Beta-constraints. For example, the Beta-constraints for   continuity are:

 

where  ,  , and   are arbitrary, but   is constrained to be positive.[8]: 65  In the case  , this reduces to   and  , for a scalar   (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   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   continuity.[citation needed]

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

Other concepts

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Relation to analyticity

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While all analytic functions are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such as bump functions (mentioned above) show that the converse is not true for functions on the reals: there exist smooth real functions that are not analytic. Simple examples of functions that are smooth but not analytic at any point can be made by means of Fourier series; another example is the Fabius function. Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered very thinly among the smooth ones; more rigorously, the analytic functions form a meagre subset of the smooth functions. Furthermore, for every open subset A of the real line, there exist smooth functions that are analytic on A and nowhere else [citation needed].

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

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

Smooth partitions of unity

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Smooth functions with given closed support are used in the construction of smooth partitions of unity (see partition of unity and topology glossary); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence. A simple case is that of a bump function on the real line, that is, a smooth function f that takes the value 0 outside an interval [a,b] and such that

 

Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals   and   to cover the whole line, such that the sum of the functions is always 1.

From what has just been said, partitions of unity do not apply to holomorphic functions; their different behavior relative to existence and analytic continuation is one of the roots of sheaf theory. In contrast, sheaves of smooth functions tend not to carry much topological information.

Smooth functions on and between manifolds

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Given a smooth manifold  , of dimension   and an atlas   then a map   is smooth on   if for all   there exists a chart   such that   and   is a smooth function from a neighborhood of   in   to   (all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any chart of the atlas that contains   since the smoothness requirements on the transition functions between charts ensure that if   is smooth near   in one chart it will be smooth near   in any other chart.

If   is a map from   to an  -dimensional manifold  , then   is smooth if, for every   there is a chart   containing   and a chart   containing   such that   and   is a smooth function from  

Smooth maps between manifolds induce linear maps between tangent spaces: for  , at each point the pushforward (or differential) maps tangent vectors at   to tangent vectors at  :   and on the level of the tangent bundle, the pushforward is a vector bundle homomorphism:   The dual to the pushforward is the pullback, which "pulls" covectors on   back to covectors on   and  -forms to  -forms:   In this way smooth functions between manifolds can transport local data, like vector fields and differential forms, from one manifold to another, or down to Euclidean space where computations like integration are well understood.

Preimages and pushforwards along smooth functions are, in general, not manifolds without additional assumptions. Preimages of regular points (that is, if the differential does not vanish on the preimage) are manifolds; this is the preimage theorem. Similarly, pushforwards along embeddings are manifolds.[10]

Smooth functions between subsets of manifolds

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There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If   is a function whose domain and range are subsets of manifolds   and   respectively.   is said to be smooth if for all   there is an open set   with   and a smooth function   such that   for all  

See also

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References

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  1. ^ Weisstein, Eric W. "Smooth Function". mathworld.wolfram.com. Archived from the original on 2019-12-16. Retrieved 2019-12-13.
  2. ^ Warner, Frank W. (1983). Foundations of Differentiable Manifolds and Lie Groups. Springer. p. 5 [Definition 1.2]. ISBN 978-0-387-90894-6. Archived from the original on 2015-10-01. Retrieved 2014-11-28.
  3. ^ Henri Cartan (1977). Cours de calcul différentiel. Paris: Hermann.
  4. ^ Barsky, Brian A. (1981). The Beta-spline: A Local Representation Based on Shape Parameters and Fundamental Geometric Measures (Ph.D.). University of Utah, Salt Lake City, Utah.
  5. ^ Brian A. Barsky (1988). Computer Graphics and Geometric Modeling Using Beta-splines. Springer-Verlag, Heidelberg. ISBN 978-3-642-72294-3.
  6. ^ Richard H. Bartels; John C. Beatty; Brian A. Barsky (1987). An Introduction to Splines for Use in Computer Graphics and Geometric Modeling. Morgan Kaufmann. Chapter 13. Parametric vs. Geometric Continuity. ISBN 978-1-55860-400-1.
  7. ^ van de Panne, Michiel (1996). "Parametric Curves". Fall 1996 Online Notes. University of Toronto, Canada. Archived from the original on 2020-11-26. Retrieved 2019-09-01.
  8. ^ a b Barsky, Brian A.; DeRose, Tony D. (1989). "Geometric Continuity of Parametric Curves: Three Equivalent Characterizations". IEEE Computer Graphics and Applications. 9 (6): 60–68. doi:10.1109/38.41470. S2CID 17893586.
  9. ^ Hartmann, Erich (2003). "Geometry and Algorithms for Computer Aided Design" (PDF). Technische Universität Darmstadt. p. 55. Archived (PDF) from the original on 2020-10-23. Retrieved 2019-08-31.
  10. ^ Guillemin, Victor; Pollack, Alan (1974). Differential Topology. Englewood Cliffs: Prentice-Hall. ISBN 0-13-212605-2.