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 (m ...
, the smoothness of a
function is a property measured by the number of
continuous derivatives
The derivative of a function is the rate of change of the function's output relative to its input value.
Derivative may also refer to:
In mathematics and economics
*Brzozowski derivative in the theory of formal languages
*Formal derivative, an ...
it has over some domain, called ''differentiability class''. 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
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
in its
domain, 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, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...
on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
and a function
defined on
with real values. Let ''k'' be a non-negative
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
. 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 se ...
expansion around any point in its domain converges to the function in some
neighborhood of the point.
is thus strictly contained in
.
Bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all ...
s are examples of functions in
but ''not'' in
.
To put it differently, the class
consists of all continuous functions. The class
consists of all
differentiable functions 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 by making precise the idea of a space having no "punctures" or "missing endpoints", ...
. Therefore,
is an example of a function that is differentiable but not locally
Lipschitz continuous.
Example: Analytic (''C'')
The
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
is analytic, and hence falls into the class ''C''
ω. 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 ...
s are also analytic wherever they are defined as they are
linear combinations of complex exponential functions and
.
Example: Smooth (''C'') but not Analytic (''C'')
The
bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all ...
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.
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, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
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, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-value ...
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
seminorms
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 by making precise the idea of a space having no "punctures" or "missing endpoints", ...
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 imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
s, it can sometimes be more fruitful to work instead with the
Sobolev spaces.
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 everyday use and 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 scalar quantity ...
, 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 defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric o ...
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