Function of a real variable

In mathematics, a function of a real variable is a function whose domain is a subset of ${\displaystyle \mathbb {R} }$. Many real functions that are often encountered have their domain contain an interval of non-empty interior, and may be continuous, or have some degree of smoothness, over one or more intervals, each of non-empty interior, in the domain. In older texts, the theory of functions of a real variable is often synonymous with what is usually now called real analysis.

The most widely considered such functions are the real functions, which are the real-valued functions 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 ${\displaystyle \mathbb {R} }$-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 ${\displaystyle \mathbb {R} }$-algebra, such as the complex numbers or the quaternions. The structure ${\displaystyle \mathbb {R} }$-vector space of the codomain induces a structure of ${\displaystyle \mathbb {R} }$-vector space on the functions. If the codomain has a structure of ${\displaystyle \mathbb {R} }$-algebra, the same is true for the functions.

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.