In mathematical analysis
, and applications in geometry
, applied mathematics
, and natural science
s, a function of a real variable is a function
is the real number
s , or a subset
of that contains an interval
of positive length. Most real functions that are considered and studied are differentiable
in some interval.
The most widely considered such functions are the real functions, which are the real-valued function
s of a real variable, that is, the functions of a real variable whose codomain
is the set of real numbers.
Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of -vector space
over the reals. That is, the codomain may be a Euclidean space
, a coordinate vector
, the set of matrices
of real numbers of a given size, or an -algebra
, such as the complex number
s or the quaternion
s. The structure -vector space of the codomain induces a structure of -vector space on the functions. If the codomain has a structure of -algebra, the same is true for the functions.
of a function of a real variable is a curve
in the codomain. In this context, a function that defines curve is called a parametric equation
of the curve.
When the codomain of a function of a real variable is a finite-dimensional vector space
, the function may be viewed as a sequence of real functions. This is often used in applications.
A real function is a function
from a subset of
denotes as usual the set of real number
s. That is, the domain
of a real function is a subset
, and its codomain
It is generally assumed that the domain contains an interval
of positive length.
For many commonly used real functions, the domain is the whole set of real numbers, and the function is continuous
at every point of the domain. One says that these functions are defined, continuous and differentiable everywhere. This is the case of:
* All polynomial function
s, including constant function
s and linear functions
* Exponential function
Some functions are defined everywhere, but not continuous at some points. For example
* The Heaviside step function
is defined everywhere, but not continuous at zero.
Some functions are defined and continuous everywhere, but not everywhere differentiable. For example
* The absolute value
is defined and continuous everywhere, and is differentiable everywhere, except for zero.
* The cubic root
is defined and continuous everywhere, and is differentiable everywhere, except for zero.
Many common functions are not defined everywhere, but are continuous and differentiable everywhere where they are defined. For example:
* A rational function
is a quotient of two polynomial functions, and is not defined at the zeros
of the denominator.
* The tangent function
is not defined for
where is any integer.
* The logarithm function
is defined only for positive values of the variable.
Some functions are continuous in their whole domain, and not differentiable at some points. This is the case of:
*The square root
is defined only for nonnegative values of the variable, and not differentiable at 0 (it is differentiable for all positive values of the variable).
A real-valued function of a real variable is a function
that takes as input a real number
, commonly represented by the variable
''x'', for producing another real number, the ''value'' of the function, commonly denoted ''f''(''x''). For simplicity, in this article a real-valued function of a real variable will be simply called a function. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified.
Some functions are defined for all real values of the variables (one says that they are everywhere defined), but some other functions are defined only if the value of the variable is taken in a subset ''X'' of ℝ, the domain
of the function, which is always supposed to contain an interval
of positive length. In other words, a real-valued function of a real variable is a function
such that its domain ''X'' is a subset of ℝ that contains an interval of positive length.
A simple example of a function in one variable could be:
which is the square root
of a function
is the set of all values of when the variable ''x'' runs in the whole domain of . For a continuous (see below for a definition) real-valued function with a connected domain, the image is either an interval
or a single value. In the latter case, the function is a constant function
of a given real number ''y'' is the set of the solutions of the equation
of a function of several real variables is a subset of ℝ that is sometimes explicitly defined. In fact, if one restricts the domain ''X'' of a function ''f'' to a subset ''Y'' ⊂ ''X'', one gets formally a different function, the ''restriction'' of ''f'' to ''Y'', which is denoted ''f''|''Y''
. In practice, it is often not harmful to identify ''f'' and ''f''|''Y''
, and to omit the subscript |''Y''
Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example by continuity
or by analytic continuation
. This means that it is not worthy to explicitly define the domain of a function of a real variable.
The arithmetic operations may be applied to the functions in the following way:
* For every real number ''r'', the constant function
, is everywhere defined.
* For every real number ''r'' and every function ''f'', the function
has the same domain as ''f'' (or is everywhere defined if ''r'' = 0).
* If ''f'' and ''g'' are two functions of respective domains ''X'' and ''Y'' such that contains an open subset of ℝ, then
are functions that have a domain containing .
It follows that the functions of ''n'' variables that are everywhere defined and the functions of ''n'' variables that are defined in some neighbourhood
of a given point both form commutative algebras
over the reals (ℝ-algebras).
One may similarly define
which is a function only if the set of the points in the domain of ''f'' such that contains an open subset of ℝ. This constraint implies that the above two algebras are not fields
Continuity and limit
Until the second part of 19th century, only continuous function
s were considered by mathematicians. At that time, the notion of continuity was elaborated for the functions of one or several real variables a rather long time before the formal definition of a topological space
and a continuous map
between topological spaces. As continuous functions of a real variable are ubiquitous in mathematics, it is worth defining this notion without reference to the general notion of continuous maps between topological space.
For defining the continuity, it is useful to consider the distance function
of ℝ, which is an everywhere defined function of 2 real variables:
A function ''f'' is continuous at a point
which is interior
to its domain, if, for every positive real number , there is a positive real number such that
In other words, may be chosen small enough for having the image by ''f'' of the interval of radius centered at
contained in the interval of length centered at
A function is continuous if it is continuous at every point of its domain.
of a real-valued function of a real variable is as follows. Let ''a'' be a point in topological closure
of the domain ''X'' of the function ''f''. The function, ''f'' has a limit ''L'' when ''x'' tends toward ''a'', denoted
if the following condition is satisfied:
For every positive real number ''ε'' > 0, there is a positive real number ''δ'' > 0 such that
for all ''x'' in the domain such that
If the limit exists, it is unique. If ''a'' is in the interior of the domain, the limit exists if and only if the function is continuous at ''a''. In this case, we have
When ''a'' is in the boundary
of the domain of ''f'', and if ''f'' has a limit at ''a'', the latter formula allows to "extend by continuity" the domain of ''f'' to ''a''.
One can collect a number of functions each of a real variable, say
into a vector parametrized by ''x'':
The derivative of the vector y is the vector derivatives of ''fi
''(''x'') for ''i'' = 1, 2, ..., ''n'':
One can also perform line integral
s along a space curve
parametrized by ''x'', with position vector
r = r(''x''), by integrating with respect to the variable ''x'':
where · is the dot product
, and ''x'' = ''a'' and ''x'' = ''b'' are the start and endpoints of the curve.
With the definitions of integration and derivatives, key theorems can be formulated, including the fundamental theorem of calculus integration by parts
, and Taylor's theorem
. Evaluating a mixture of integrals and derivatives can be done by using theorem differentiation under the integral sign
A real-valued implicit function
of a real variable is not written in the form "''y'' = ''f''(''x'')". Instead, the mapping is from the space ℝ2
to the zero element
in ℝ (just the ordinary zero 0):
is an equation in the variables. Implicit functions are a more general way to represent functions, since if:
then we can always define:
but the converse is not always possible, i.e. not all implicit functions have the form of this equation.
One-dimensional space curves in ℝ''n''
Given the functions , , ..., all of a common variable ''t'', so that:
or taken together:
then the parametrized ''n''-tuple,
describes a one-dimensional space curve
Tangent line to curve
At a point for some constant ''t'' = ''c'', the equations of the one-dimensional tangent line to the curve at that point are given in terms of the ordinary derivative
s of ''r''1
(''t''), ..., ''r''''n''
(''t''), and ''r'' with respect to ''t'':
Normal plane to curve
The equation of the ''n''-dimensional hyperplane normal to the tangent line at r = a is:
or in terms of the dot product
where are points ''in the plane'', not on the space curve.
Relation to kinematics
The physical and geometric interpretation of ''d''r(''t'')/''dt'' is the "velocity
" of a point-like particle
moving along the path r(''t''), treating r as the spatial position vector
coordinates parametrized by time ''t'', and is a vector tangent to the space curve for all ''t'' in the instantaneous direction of motion. At ''t'' = ''c'', the space curve has a tangent vector , and the hyperplane normal to the space curve at ''t'' = ''c'' is also normal to the tangent at ''t'' = ''c''. Any vector in this plane (p − a) must be normal to .
is the "acceleration
" of the particle, and is a vector normal to the curve directed along the radius of curvature
Matrix valued functions
can also be a function of a single variable. For example, the rotation matrix
is a matrix valued function of rotation angle of about the origin. Similarly, in special relativity
, the Lorentz transformation
matrix for a pure boost (without rotations):
is a function of the boost parameter ''β'' = ''v''/''c'', in which ''v'' is the relative velocity
between the frames of reference (a continuous variable), and ''c'' is the speed of light
, a constant.
Banach and Hilbert spaces and quantum mechanics
Generalizing the previous section, the output of a function of a real variable can also lie in a Banach space or a Hilbert space. In these spaces, division and multiplication and limits are all defined, so notions such as derivative and integral still apply. This occurs especially often in quantum mechanics, where one takes the derivative of a ket
or an operator
. This occurs, for instance, in the general time-dependent Schrödinger equation
where one takes the derivative of a wave function, which can be an element of several different Hilbert spaces.
Complex-valued function of a real variable
A complex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex
If is such a complex valued function, it may be decomposed as
: = + ,
where and are real-valued functions. In other words, the study of the complex valued functions reduces easily to the study of the pairs of real valued functions.
Cardinality of sets of functions of a real variable
of the set of real-valued functions of a real variable,
, which is strictly larger than the cardinality of the continuum
(i.e., set of all real numbers). This fact is easily verified by cardinal arithmetic:
is a set such that
, then the cardinality of the set
However, the set of continuous function
has a strictly smaller cardinality, the cardinality of the continuum,
. This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain.
Thus, the cardinality of the set of continuous real-valued functions on the reals is no greater than the cardinality of the set of real-valued functions of a rational variable. By cardinal arithmetic:
On the other hand, since there is a clear bijection between
and the set of constant functions
, which forms a subset of
must also hold. Hence,
*Function of several real variables
*Several complex variables
External links''Multivariable Calculus''L. A. Talman (2007) ''Differentiability for Multivariable Functions''