Function of a real variable

The image 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.

For many commonly used real functions, the domain is the whole set of real numbers, and the function is continuous and differentiable at every point of the domain. One says that these functions are defined, continuous and differentiable everywhere. This is the case of:

Some functions are defined everywhere, but not continuous at some points. For example

Some functions are defined and continuous everywhere, but not everywhere differentiable. For example

Many common functions are not defined everywhere, but are continuous and differentiable everywhere where they are defined. For example:

Some functions are continuous in their whole domain, and not differentiable at some points. This is the case of:

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.

The preimage of a given real number y is the set of the solutions of the equation y = f(x).

The domain 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 YX, 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:

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).

Until the second part of 19th century, only continuous functions 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.

The limit of a real-valued function of a real variable is as follows.[1] 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

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.

The derivative of the vector y is the vector derivatives of fi(x) for i = 1, 2, ..., n:

One can also perform line integrals 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:

but the converse is not always possible, i.e. not all implicit functions have the form of this equation.

Given the functions r1 = r1(t), r2 = r2(t), ..., rn = rn(t) all of a common variable t, so that:

At a point r(t = c) = a = (a1, a2, ..., an) 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 derivatives of r1(t), r2(t), ..., rn(t), and r with respect to t:

The equation of the n-dimensional hyperplane normal to the tangent line at r = a is:

where p = (p1, p2, ..., pn) are points in the plane, not on the space curve.

The physical and geometric interpretation of dr(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 dr(t)/dt|t = c, 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 (pa) must be normal to dr(t)/dt|t = c.

Similarly, d2r(t)/dt2 is the "acceleration" of the particle, and is a vector normal to the curve directed along the radius of curvature.

A matrix can also be a function of a single variable. For example, the rotation matrix in 2d:

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.

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.

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 values.

where g and h 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.