In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
, the smoothness of a
function 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 g ...
derivatives 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
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
* ...
, 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 are su ...
on the
real line 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 languag ...
. The function
is said to be of differentiability class ''
'' if the derivatives
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 g ...
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
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
, if
is smooth (i.e.,
is in the class
) and its
Taylor series expansion around any point in its domain converges to the function in some
neighborhood of the point.
is thus strictly contained in
.
Bump functions 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 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. 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 a ...
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
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
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 a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
s
where
varies over an increasing sequence of
compact sets 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 equations, 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 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 quant ...
, with which the parameter traces out the curve.
Parametric continuity
Parametric continuity (''C''
''k'') is a concept applied to
parametric curves, 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