In
mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, the smoothness of a
function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
is a property measured by the number of
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...
derivatives
Derivative may refer to:
In mathematics and economics
*Brzozowski derivative in the theory of formal languages
*Derivative in calculus, a quantity indicating how a function changes when the values of its inputs change.
*Formal derivative, an opera ...
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
Orders is a surname
In some cultures, a surname, family name, or last name is the portion of one's personal name that indicates their family, tribe or community.
Practices vary by culture. The family name may be placed at either the start of ...
in its
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...
, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or
function).
Differentiability classes
Differentiability class is a classification of functions according to the properties of their
derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s. It is a measure of the highest order of derivative that exists for a function.
Consider an
open set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
on the
real line
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and a function ''f'' defined on that set with real values. Let ''k'' be a non-negative
integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
. The function ''f'' is said to be of (differentiability) class ''C
k'' if the derivatives ''f''′, ''f''″, ..., ''f''
(''k'') exist and are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...
. 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''
ω, or
analytic, if ''f'' is smooth ''and'' if its
Taylor series
In , the Taylor series of a is an of terms that are expressed in terms of the function's s at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after ...
expansion around any point in its domain converges to the function in some neighborhood of the point. ''C''
ω is thus strictly contained in ''C''
∞.
Bump function
In mathematics, a bump function (also called a test function) is a Function (mathematics), function f: \mathbf^n \to \mathbf on a Euclidean space \mathbf^n which is both smooth function, smooth (in the sense of having Continuous function, continuo ...

s are examples of functions in ''C''
∞ but ''not'' in ''C''
ω.
To put it differently, the class ''C''
0 consists of all continuous functions. The class ''C''
1 consists of all
differentiable function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s whose derivative is continuous; such functions are called continuously differentiable. Thus, a ''C''
1 function is exactly a function whose derivative exists and is of class ''C''
0. In general, the classes ''C
k'' can be defined
recursively
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...

by declaring ''C''
0 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'' > 0, and there are examples to show that this containment is strict (''C
k'' ⊊ ''C''
''k''−1). The class ''C''
∞ of infinitely differentiable functions, is the intersection of the classes ''C
k'' as ''k'' varies over the non-negative integers.
Examples

The function
is continuous, but not differentiable at , so it is of class ''C''
0, but not of class ''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. Moreover, if one takes
( ≠ 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
In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed set, closed (i.e., containing all its limit points) and bounded set, bounded (i.e., having all ...
and, therefore, that a differentiable function on a compact set may not be locally
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ex ...
.
The functions
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
The exponential function is a mathematical function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of a ...

is analytic, and hence falls into the class ''C''
ω. The
trigonometric function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s are also analytic wherever they are defined.
The
bump function
In mathematics, a bump function (also called a test function) is a Function (mathematics), function f: \mathbf^n \to \mathbf on a Euclidean space \mathbf^n which is both smooth function, smooth (in the sense of having Continuous function, continuo ...

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
Compact as used in politics may refer broadly to a pact
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations, pacts are usually between two or more sovereign state
A sovereign state is a po ...
.
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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
exist and are continuous, for every
non-negative integers, such that
, and every
. Equivalently,
is of class
on
if the
-th order
Fréchet derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of
exists and is continuous at every point of
. The function
is said to be of class
or
if it is continuous on
.
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 ''D'' be an open subset of the real line. The set of all ''C
k'' real-valued functions defined on ''D'' is a
Fréchet vector space, with the countable family of
seminorm In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
s
where ''K'' varies over an increasing sequence of
compact set
In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed set, closed (i.e., containing all its limit points) and bounded set, bounded (i.e., having all ...
s 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 equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 normed space, norm that is a combination of Lp norm, ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a s ...
s.
Parametric continuity
The terms ''parametric continuity'' 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 everyday use and in kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

, with which the parameter traces out the curve.
Parametric continuity is a concept applied to
parametric curve
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, which describes the smoothness of the parameter's value with distance along the curve.
Definition
A (parametric) curve
is said to be of class ''C''
''k'', if
exists and is continuous on