Editing Function of a real variable (section)

{{Short description|Mathematical function}}
{{functions}}

In [[mathematical analysis]], and applications in [[geometry]], [[applied mathematics]], [[engineering]], and [[natural science]]s, a '''function of a real variable''' is a [[function (mathematics)|function]] whose [[domain of a function|domain]] is the [[real number]]s <math>\mathbb{R}</math>, or a [[subset]] of <math>\mathbb{R}</math> that contains an [[interval (mathematics)|interval]] of positive length. Most real functions that are considered and studied are [[differentiable function|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 <math>\mathbb{R}</math>-[[vector space]] over the reals. That is, the codomain may be a [[Euclidean space]], a [[coordinate vector]], the set of [[matrix (mathematics)|matrices]] of real numbers of a given size, or an <math>\mathbb{R}</math>-[[algebra over a field|algebra]], such as the [[complex number]]s or the [[quaternion]]s. The structure <math>\mathbb{R}</math>-vector space of the codomain induces a structure of <math>\mathbb{R}</math>-vector space on the functions. If the codomain has a structure of <math>\mathbb{R}</math>-algebra, the same is true for the functions.

The [[image (mathematics)|image]] of a function of a real variable is a [[curve (mathematics)|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.