In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, the smoothness of a
function is a property measured by the number of
continuous derivatives (''differentiability class)'' it has over its
domain.
A function of class
is a function of smoothness at least ; that is, a function of class
is a function that has a th 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
Differentiability class is a classification of functions according to the properties of their
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s. It is a measure of the highest order of derivative that exists and is continuous for a function.
Consider an
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
on the
real line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
and a function
defined on
with real values. Let ''k'' be a non-negative
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
. 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
)
The function
is said to be of class
or ''
analytic'', if
is smooth (i.e.,
is in the class
) and its
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion around any point in its domain converges to the function in some
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of the point. There exist functions that are smooth but not analytic;
is thus strictly contained in
Bump function
In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supp ...
s are examples of functions with this property.
To put it differently, the class
consists of all continuous functions. The class
consists of all
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
s 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
Example: continuous (''C''0) but not differentiable

The function
is continuous, but not differentiable at , so it is of class ''C''
0, but not of class ''C''
1.
Example: finitely-times differentiable (''C'')
For each even integer , the function
is continuous and times differentiable at all . At , however,
is not times differentiable, so
is of class ''C''
, but not of class ''C''
where .
Example: differentiable but not continuously differentiable (not ''C''1)
The function
is differentiable, with derivative
Because
oscillates as → 0,
is not continuous at zero. Therefore,
is differentiable but not of class ''C''
1.
Example: differentiable but not Lipschitz continuous
The function
is differentiable but its derivative is unbounded on a
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
. Therefore,
is an example of a function that is differentiable but not locally
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after Germany, German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in h ...
.
Example: analytic (''C'')
The
exponential function is
analytic, and hence falls into the class ''C''
ω (where ω is the smallest
transfinite ordinal
In mathematics, transfinite numbers or infinite numbers are numbers that are " infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of in ...
). The
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s are also analytic wherever they are defined, because they are
linear combinations of complex exponential functions and
.
Example: smooth (''C'') but not analytic (''C'')
The
bump function
In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supp ...
is smooth, so of class ''C''
∞, but it is not analytic at , and hence is not of class ''C''
ω. The function is an example of a smooth function with
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed ...
.
Multivariate differentiability classes
A function
defined on an open set
of
is said 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 ''C''''k'' functions
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
seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
s
where
varies over an increasing sequence of
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
s 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 equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s, it can sometimes be more fruitful to work instead with the
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s.
Continuity
The terms ''parametric continuity'' (''C''
''k'') and ''geometric continuity'' (''G
n'') were introduced by
Brian Barsky, to show that the smoothness of a curve could be measured by removing restrictions on the
speed
In kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. Intro ...
, with which the parameter traces out the curve.
Parametric continuity
Parametric continuity (''C''
''k'') is a concept applied to
parametric curve
In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters.
In the case ...
s, which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve
is said to be of class ''C''
''k'', if
exists and is continuous on
1 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:
*
C^0: zeroth derivative is continuous (curves are continuous)
*
C^1: zeroth and first derivatives are continuous
*
C^2: zeroth, first and second derivatives are continuous
*
C^n: 0-th through
n-th derivatives are continuous
Geometric continuity

A
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
or
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
can be described as having
G^n continuity, with
n being the increasing measure of smoothness. Consider the segments either side of a point on a curve:
*
G^0: The curves touch at the join point.
*
G^1: The curves also share a common
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
direction at the join point.
*
G^2: The curves also share a common center of curvature at the join point.
In general,
G^n continuity exists if the curves can be reparameterized to have
C^n (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) such that
f(1)=g(0) have
G^n continuity at the point where they meet if
they satisfy equations known as Beta-constraints. For example, the Beta-constraints for
G^4 continuity are:
:
\begin
g^(0) & = \beta_1 f^(1) \\
g^(0) & = \beta_1^2 f^(1) + \beta_2 f^(1) \\
g^(0) & = \beta_1^3 f^(1) + 3\beta_1\beta_2 f^(1) +\beta_3 f^(1) \\
g^(0) & = \beta_1^4 f^(1) + 6\beta_1^2\beta_2 f^(1) +(4\beta_1\beta_3+3\beta_2^2) f^(1) +\beta_4 f^(1) \\
\end
where
\beta_2,
\beta_3, and
\beta_4 are arbitrary, but
\beta_1 is constrained to be positive.
In the case
n=1, this reduces to
f'(1)\neq0 and
f'(1) = kg'(0), for a scalar
k>0 (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
G^1 continuity to appear smooth, for good
aesthetics
Aesthetics (also spelled esthetics) is the branch of philosophy concerned with the nature of beauty and taste (sociology), taste, which in a broad sense incorporates the philosophy of art.Slater, B. H.Aesthetics ''Internet Encyclopedia of Ph ...
, such as those aspired to in
architecture
Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and construction, constructi ...
and
sports car
A sports car is a type of automobile that is designed with an emphasis on dynamic performance, such as Automobile handling, handling, acceleration, top speed, the thrill of driving, and Auto racing, racing capability. Sports cars originated in ...
design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has
G^2 continuity.
A (with ninety degree circular arcs at the four corners) has
G^1 continuity, but does not have
G^2 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^2 continuity is required, then
cubic splines are typically chosen; these curves are frequently used in
industrial design
Industrial design is a process of design applied to physical Product (business), products that are to be manufactured by mass production. It is the creative act of determining and defining a product's form and features, which takes place in adva ...
.
Other concepts
Relation to analyticity
While all
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such as
bump function
In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supp ...
s (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
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
; 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 number
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . ...
s 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
In mathematics, a partition of unity on a topological space is a Set (mathematics), set of continuous function (topology), continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood (mathem ...
'' and
topology glossary); these are essential in the study of
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
s, for example to show that
Riemannian metric
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s can be defined globally starting from their local existence. A simple case is that of a ''
bump function
In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supp ...
'' 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 \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
(-\infty, c] 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 function">, +\infty) 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 (mathematics), 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
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
M, of dimension
m, and an
atlas
An atlas is a collection of maps; it is typically a bundle of world map, maps of Earth or of a continent or region of Earth. Advances in astronomy have also resulted in atlases of the celestial sphere or of other planets.
Atlases have traditio ...
\mathfrak = \_\alpha, then a map
f:M\to \R is smooth on
M if for all
p \in M there exists a chart
(U, \phi) \in \mathfrak, such that
p \in U, and
f \circ \phi^ : \phi(U) \to \R is a smooth function from a neighborhood of
\phi(p) in
\R^m to
\R (all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any
chart
A chart (sometimes known as a graph) is a graphics, graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can repres ...
of the atlas that contains
p, since the smoothness requirements on the transition functions between charts ensure that if
f is smooth near
p in one chart it will be smooth near
p in any other chart.
If
F : M \to N is a map from
M to an
n-dimensional manifold
N, then
F is smooth if, for every
p \in M, there is a chart
(U,\phi) containing
p, and a chart
(V, \psi) containing
F(p) such that
F(U) \subset V, and
\psi \circ F \circ \phi^ : \phi(U) \to \psi(V) is a smooth function from
\R^n.
Smooth maps between manifolds induce linear maps between
tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
s: for
F : M \to N, at each point the
pushforward (or differential) maps tangent vectors at
p to tangent vectors at
F(p):
F_ : T_p M \to T_N, and on the level of the
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
, the pushforward is a
vector bundle homomorphism:
F_* : TM \to TN. The dual to the pushforward is the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
, which "pulls" covectors on
N back to covectors on
M, and
k-forms to
k-forms:
F^* : \Omega^k(N) \to \Omega^k(M). In this way smooth functions between manifolds can transport
local data, like
vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s and
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
s, 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
f : X \to Y is a
function whose
domain and
range are subsets of manifolds
X \subseteq M and
Y \subseteq N respectively.
f is said to be smooth if for all
x \in X there is an open set
U \subseteq M with
x \in U and a smooth function
F : U \to N such that
F(p) = f(p) for all
p \in U \cap X.
See also
*
*
*
*
*
*
*
* (number theory)
*
*
*
Sobolev mapping
References
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