Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. 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 in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or formula_1 function). Differentiability classes. 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 for a function. Consider an open set on the real line and a function "f" defined on that set with real values. Let "k" be a non-negative integer. The function "f" is said to be of (differentiability) class "C" if the derivatives "f"′, "f"″, ..., "f" exist and are continuous. The function "f" is said to be infinitely differentiable, smooth, or of class "C", if it has derivatives of all orders. The function "f" is said to be of class "C", or analytic, if "f" is smooth "and" if its Taylor series expansion around any  point in its domain converges to the function in some neighborhood of the point. "C" is thus strictly contained in "C". Bump functions are examples of functions in "C" but "not" in "C". To put it differently, the class "C" consists of all continuous functions. The class "C" consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a "C" function is exactly a function whose derivative exists and is of class "C". In general, the classes "C" can be defined recursively by declaring "C" to be the set of all continuous functions, and  declaring "C" for any positive integer "k" to be the set of all differentiable functions whose derivative is in "C". In particular, "C" is contained in "C" for every "k" > 0, and there are examples to show that this containment is strict ("C" ⊊ "C"). The class "C" of infinitely differentiable functions, is the intersection of the classes "C" as "k" varies over the non-negative integers. Examples. The function formula_2 is continuous, but not differentiable at , so it is of class "C", but not of class "C". The function formula_3 is differentiable, with derivative formula_4 Because formula_5 oscillates  as → 0, formula_6 is not continuous at zero. Therefore, formula_7 is differentiable but not of class "C". Moreover, if one takes formula_8 ( ≠ 0) in this example, it can be used to show that the derivative function of a differentiable function can be unbounded on a compact set and, therefore, that a differentiable function on a compact set may not be locally Lipschitz continuous. The functions formula_9 where is even, are continuous and times differentiable at all . But at they are not times differentiable, so they are of class "C", but not of class "C" where . The exponential function  is analytic, and hence falls into the class "C". The trigonometric functions are also analytic wherever they are defined. The bump function formula_10 is smooth, so of class "C", but it is not analytic at , and hence is not of class "C". The function is an example of a smooth function with compact support. Multivariate differentiability classes. A function formula_11 defined on an open set formula_12 of formula_13 is said to be of class formula_14 on formula_12, for a positive integer formula_16, if all partial derivatives formula_17 exist and are continuous, for every formula_18 non-negative integers, such that formula_19 , and every formula_20. Equivalently, formula_21 is of class formula_14 on formula_12 if the formula_16-th order Fréchet derivative of formula_21 exists and is continuous at every point of formula_12. The function formula_21 is said to be of class formula_28 or formula_29 if it is continuous on formula_12. A function formula_31, defined on an open set formula_12 of formula_13, is said to be of class formula_14 on formula_12, for a positive integer formula_16, if all of its components formula_37 are of class formula_14, where formula_39 are the natural projections formula_40 defined by formula_41. It is said to be of class formula_28  or formula_29 if it is continuous, or equivalently, if all components formula_44 are continuous, on formula_12. The space of "C" functions. Let "D" be an open subset of the real line. The set of all "C" real-valued functions defined on "D" is a Fréchet vector space, with the countable family of seminorms formula_46 where "K" varies over an increasing sequence of compact sets whose union is "D", and "m" = 0, 1, ..., "k". The set of "C" 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 spaces. Continuity. The terms "parametric continuity" ("C") and "geometric continuity" ("G") 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. Parametric continuity. Parametric continuity (C) is a concept applied to parametric curves, which describes the smoothness  of the parameter's value with distance along the curve. A (parametric) curve formula_47 is said to be of class "C", if formula_48 exists and is continuous on formula_49, where derivatives at the end-points formula_50 are taken to be one sided derivatives (i.e., at formula_51 from the right, and at formula_52 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" 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: Geometric continuity. The concept of geometrical continuity or geometric continuity (G) was primarily applied to the conic sections (and related shapes) by mathematicians such as Leibniz, Kepler, 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 tends to a circle as the eccentricity approaches zero, or to a parabola as it approaches one; and a hyperbola tends to a parabola 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. For such to be the case, one would have to make the line closed  by allowing the point formula_53 to be a point on the circle, and for formula_54 and formula_55 to be identical. Such ideas were useful in crafting the modern, algebraically defined, idea of the continuity of a function and of formula_56 (see projectively extended real line for more). Order of geometric continuity. A curve or surface can be described as having "G" continuity, with "n" being the increasing measure of smoothness. Consider the segments either side of a point on a curve: In general, "G" continuity exists if the curves can be reparameterized to have "C" (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" continuity if and , for a scalar (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" 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 "G" continuity. A (with ninety degree  circular arcs at the four corners) has "G" continuity, but does not have "G" 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" 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 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 . 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 . Smooth partitions of unity. 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 formula_57 Given a number of overlapping intervals on the line, bump functions can be  constructed on each of them, and on semi-infinite intervals formula_58 and formula_59 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 theory. In contrast, sheaves of smooth functions tend not to carry much topological information. Smooth functions on and between manifolds. Given a smooth manifold formula_60, of dimension formula_61 and an atlas formula_62 then a map formula_63 is smooth on formula_60 if  for all formula_65 there exists a chart formula_66 such that formula_67 and formula_68 is a smooth function from a neighborhood of formula_69 in formula_70 to formula_71 (all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any chart of the atlas that contains formula_72 since the smoothness requirements on the transition functions between charts ensure that if formula_21is smooth near formula_74 in one chart it will be smooth near formula_74 in any other chart. If formula_76 is a map from formula_60 to an formula_78-dimensional manifold formula_79, then formula_80 is smooth if, for  every formula_81 there is a chart formula_82 containing formula_72 and a chart formula_84 containing formula_85 such that formula_86 and formula_87 is a smooth function from formula_88 Smooth maps between manifolds induce linear maps between tangent spaces: for formula_76, at each point the pushforward (or differential) maps tangent vectors at formula_74 to tangent vectors at formula_85: formula_92 and on the level of the tangent bundle, the pushforward is a vector bundle homomorphism: formula_93 The dual to the pushforward is the pullback, which "pulls" covectors on formula_79 back to covectors on formula_95 and formula_16-forms to formula_16-forms: formula_98 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. Smooth functions between subsets of manifolds. There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If formula_99 is a function whose domain and  range are subsets of manifolds formula_100 and formula_101 respectively. formula_21 is said to be smooth if for all formula_103 there is an open set formula_104 with formula_105 and a smooth function formula_106 such that formula_107 for all formula_108