In mathematics, a function of a real variable is a function whose domain is a subset of R {\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 R {\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 R {\displaystyle \mathbb {R} } -algebra, such as the complex numbers or the quaternions. The structure R {\displaystyle \mathbb {R} } -vector space of the codomain induces a structure of R {\displaystyle \mathbb {R} } -vector space on the functions. If the codomain has a structure of R {\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.
Real function
A real function is a function from a subset of R {\displaystyle \mathbb {R} } to R , {\displaystyle \mathbb {R} ,} where R {\displaystyle \mathbb {R} } denotes as usual the set of real numbers. That is, the domain of a real function is a subset R {\displaystyle \mathbb {R} } , and its codomain is R . {\displaystyle \mathbb {R} .} It is generally assumed that the domain contains an interval of positive length.
Basic examples 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:
All polynomial functions, including constant functions and linear functions Sine and cosine 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 π 2 + k π , {\displaystyle {\frac {\pi }{2}}+k\pi ,} where k 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).
General definition 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 {\displaystyle \mathbb {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
f : X → R {\displaystyle f:X\to \mathbb {R} }
such that its domain X is a subset of R {\displaystyle \mathbb {R} } that contains an interval of positive length. A simple example of a function in one variable could be:
f : X → R {\displaystyle f:X\to \mathbb {R} }
X = { x ∈ R : x ≥ 0 } {\displaystyle X=\{x\in \mathbb {R} \,:\,x\geq 0\}}
f ( x ) = x {\displaystyle f(x)={\sqrt {x}}}
which is the square root of x.
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