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Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.

Most of this article will be about real-valued functions of one real variable. A discussion of the multivariable case will be presented towards the end.

Differentiability classes

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 class Ck if the derivatives f', f'', ..., f(k) exist and are continuous (the continuity is automatic for all the derivatives except the last one, f(k)). The function f is said to be of class C, or smooth, if it has derivatives of all orders. f is said to be of class Cω, or analytic, if f is smooth and if it equals its Taylor series expansion around any point in its domain.

To put it differently, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C1 function is exactly a function whose derivative exists and is of class C0. In general, the classes Ck can be defined recursively by declaring C0 to be the set of all continuous functions and declaring Ck for any positive integer k to be the set of all differentiable functions whose derivative is in Ck-1. In particular, Ck-1 is contained in Ck for every k, and there are examples to show that this containment is strict. C is the intersection of the sets Ck as k varies over the non-negative integers. Cω is strictly contained in C; for an example of this, see bump function or also below.

Examples

The function

f(x) = begin{cases}x & mbox{if }x ge 0, 0 &mbox{if }x < 0end{cases}

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

The function

f(x) = begin{cases}x^2sin{(1/x)} & mbox{if }x neq 0, 0 &mbox{if }x = 0end{cases}
is differentiable, with derivative
f'(x) = begin{cases}2xsin{(1/x)} - cos{(1/x)} & mbox{if }x neq 0, 0 &mbox{if }x = 0.end{cases}
Because cos(1/x) oscillates as x approaches zero, f ’(x) is not continuous at zero. Therefore, this function is differentiable but not of class C1. Moreover, if one takes f(x)=x3/2sin(1/x) (x ≠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 exponential function is analytic, so, of class Cω. The trigonometric functions are also analytic wherever they are defined.

The function

f(x) = begin{cases}e^{-1/(1-x^2)} & mbox{ if } |x| < 1, 0 &mbox{ otherwise }end{cases}

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

Relation to analyticity

While all analytic functions are smooth on the set on which they are analytic, the above example shows that the converse is not true for functions on the reals: there exist smooth real functions which are not analytic. 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 which 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.

The space of Ck functions

Let D be an open subset of the real line. The set of all Ck functions defined on D and taking real values is a Fréchet space with the countable family of seminorms

p_{K, m}=sup_{xin K}left|f^{(m)}(x)right|

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.

Differentiability classes in several variables

Let n and m be some positive integers. If f is a function from an open subset of Rn with values in Rm, then f has component functions f1, ..., fm. Each of these may or may not have partial derivatives. We say that f is of class Ck if all of the partial derivatives scriptstyle partial^k f/partial x_{i_1},partial x_{i_2},cdots,partial x_{i_k} exist and are continuous, where each of scriptstyle i_1, i_2, ldots, i_k is an integer between 1 and n. The classes C and Cω are defined as before.

These criteria of differentiability can be applied to the transition functions of a differential structure. The resulting space is called a Ck manifold.

If one wishes to start with a coordinate independent definition of the class Ck, one may start by considering maps between Banach spaces. A map from one Banach space to another is differentiable at a point if there is an affine map which approximates it at that point. The derivative of the map assigns to the point x the linear part of the affine approximation to the map at x. Since the space of linear maps from one Banach space to another is again a Banach space, we may continue this procedure to define higher order derivatives. A map f is of class Ck if it has continuous derivatives up to order k, as before.

Note that Rn is a Banach space for any value of n, so the coordinate free approach is applicable in this instance. It can be shown that the definition in terms of partial derivatives and the coordinate free approach are equivalent; that is, a function f is of class Ck by one definition iff it is so by the other definition.

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

f(x) > 0 for a < x < b.

Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals (-∞, c] and [d,+∞) 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 between manifolds

Smooth maps between smooth manifolds may be defined by means of charts, since the idea of smoothness of function is independent of the particular chart used. Such a map has a first derivative defined on tangent vectors; it gives a fibre-wise linear mapping on the level of tangent bundles.

See also

References

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