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.
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.
is continuous, but not differentiable at , so it is of class C0 but not of class C1.
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 set of C∞ 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.
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 exist and are continuous, where each of is an integer between 1 and n. The classes C∞ and Cω are defined as before.
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 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 (-∞, 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 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.
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