Smooth function

A bump function is a smooth function with compact support.

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 is about real-valued functions of one real variable. A discussion of the multivariable case is 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 C^k if the derivatives f', f'', ..., f^(k) exist and are continuous (the continuity is automatic for all the derivatives except for f^(k)). The function f is said to be of class C^∞, or smooth, if it has derivatives of all orders.^[1] f is said to be of class C^ω, 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 C^k can be defined recursively by declaring C⁰ to be the set of all continuous functions and declaring C^k for any positive integer k to be the set of all differentiable functions whose derivative is in C^k−1. In particular, C^k is contained in C^k−1 for every k, and there are examples to show that this containment is strict. C^∞ is the intersection of the sets C^k as k varies over the non-negative integers. C^ω is strictly contained in C^∞; for an example of this, see bump function or also below.

Examples

The C⁰ function f(x)=x for x≥0 and 0 otherwise.

The function f(x)=x² sin(1/x) for x>0.

A smooth function that is not analytic.

The function

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

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

The function

$f(x) = \begin{cases}x^2\sin{(1/x)} & \mbox{if }x \neq 0, \\ 0 &\mbox{if }x = 0\end{cases}$

is differentiable, with derivative

$f'(x) = \begin{cases}-\mathord{\cos(1/x)} + 2x\sin{(1/x)} & \mbox{if }x \neq 0, \\ 0 &\mbox{if }x = 0.\end{cases}$

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) = x^3/2 sin(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 functions

f(x) = | x | ^{k + 1}

where k is even, are continuous and k times differentiable at all x. But at $\scriptstyle x=0$ they are not (k+1) times differentiable, so they are of class C ^k but not of class C ^j where j>k.

The exponential function is analytic, so, of class C^ω. 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 $\scriptstyle x=\pm 1$, so it is not of class C^ω. The function f is an example of a smooth function with compact support.

Multivariate differentiability classes

Let n and m be some positive integers. If f is a function from an open subset of Rⁿ with values in R^m, then f has component functions f₁, ..., f_m. Each of these may or may not have partial derivatives. We say that f is of class C^k if all of the partial derivatives $\frac{\partial^l f_i}{\partial x_{i_1}^{l_1}\partial x_{i_2}^{l_2}\cdots\partial x_{i_n}^{l_n}}$ exist and are continuous, where each of $i, i_1, i_2, \ldots, i_k$ is an integer between 1 and n, each of $l, l_1, l_2, \ldots l_n$ is an integer between 0 and k, $l_1+l_2+\cdots +l_n=l$.^[1] The classes C^∞ and C^ω are defined as before.^[1]

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

If one wishes to start with a coordinate-independent definition of the class C^k, 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 C^k 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 C^k by one definition iff it is so by the other definition.

The space of C^k functions

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

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

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 $\scriptstyle D$ also forms a Fréchet space. One uses the same seminorms as above, except that $\scriptstyle m$ 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.

Parametric continuity

Parametric continuity is a concept applied to parametric curves describing the smoothness of the parameter's value with distance along the curve.

Definition

A curve can be said to have Cⁿ continuity if

$\frac{d^n s}{dt^n}$

is continuous of value throughout the curve.

As an example of a practical application of this concept, a curve describing the motion of an object with a parameter of time, must have C¹ continuity for the object to have finite acceleration. For smoother motion, such as that of a camera's path while making a film, higher levels of parametric continuity are required.

Order of continuity

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:^[2]

• C⁻¹: curves include discontinuities
• C⁰: curves are joined
• C¹: first derivatives are equal
• C²: first and second derivatives are equal
• Cⁿ: first through n^th derivatives are equal

The term parametric continuity was introduced to distinguish it from geometric continuity (Gⁿ) which removes restrictions on the speed with which the parameter traces out the curve.^[3]

Geometric continuity

The concept of geometrical or geometric continuity 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 x = ∞ to be a point on the circle, and for x = +∞ and x = −∞ to be identical. Such ideas were useful in crafting the modern, algebraically defined, idea of the continuity of a function and of ∞.

Smoothness of curves and surfaces

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

• G⁰: The curves touch at the join point.
• G¹: The curves also share a common tangent direction at the join point.
• G²: The curves also share a common center of curvature at the join point.

In general, Gⁿ continuity exists if the curves can be reparameterized to have Cⁿ (parametric) continuity.^[4] A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.

Equivalently, two vector functions $\scriptstyle f(s)$ and $\scriptstyle g(t)$ have Gⁿ continuity if $\scriptstyle f^{(n)}(t) \,\neq\, 0$ and $\scriptstyle f^{(n)}(t) \,=\, kg^{(n)}(t)$, for a scalar $\scriptstyle k \,>\, 0$ (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 G¹ 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 G² continuity.^{[citation needed]}

A rounded rectangle (with ninety degree circular arcs at the four corners) has G¹ continuity, but does not have G² 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 G² continuity is required, then cubic splines are typically chosen; these curves are frequently used in industrial design.

Smoothness

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. For example, the Fabius function is smooth but not analytic at any point. 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.

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

$f(x) > 0 \quad \text{ for } \quad a < x < b.\,$

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. If F is a map from an m-manifold M to an n-manifold N, then F is smooth if, for every $\scriptstyle p \in M$, there is a chart $\scriptstyle (U, \varphi)$ in M containing p and a chart $\scriptstyle (V, \psi)$ in N containing F(p) with $\scriptstyle F(U) \subset V$, such that $\scriptstyle\psi\circ F \circ \varphi^{-1}$ is smooth from $\scriptstyle\varphi(U)$ to $\scriptstyle\psi(V)$ as a function from $\scriptstyle\mathbb{R}^m$ to $\scriptstyle\mathbb{R}^n$.

Such a map has a first derivative defined on tangent vectors; it gives a fibre-wise linear mapping on the level of tangent bundles.

Smooth functions between subsets of manifolds

There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If $\scriptstyle f : X \to Y$ is a function whose domain and range are subsets of manifolds $\scriptstyle X \subset M$ and $\scriptstyle Y \subset N$ respectively. $\scriptstyle f$ is said to be smooth if for all $\scriptstyle x \in X$ there is an open set $\scriptstyle U \subset M$ with $\scriptstyle x \in U$ and a smooth function $\scriptstyle F : U \to N$ such that $\scriptstyle F(p)=f(p)$ for all $\scriptstyle p \in U \cap X$.

See also

• Non-analytic smooth function
• Quasi-analytic function

References

1. ^ ^{a b c} Warner (1883), p. 5, Definition 1.2.
2. ^ Parametric Curves
3. ^ (Bartels, Beatty & Barsky 1987, Ch. 13)
4. ^ Brian A. Barsky and Tony D. DeRose, "Geometric Continuity of Parametric Curves: Three Equivalent Characterizations," IEEE Computer Graphics and Applications, 9(6), Nov. 1989, pp. 60–68.

•  This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed (1911). Encyclopædia Britannica (11th ed.). Cambridge University Press. 
• Guillemin, Pollack. Differential Topology. Prentice-Hall (1974).
• Warner, Frank Wilson (1983). Foundations of differentiable manifolds and Lie groups. Springer. ISBN 9780387908946.