Function of a real variable
Encyclopedia
In mathematics
, a function of a real variable is a mathematical function whose domain
is the real line
. More loosely, a function of a real variable is sometimes taken to mean any function whose domain is a subset of the real line
.
Functions of a real variable were the classical object of study in mathematical analysis
, specifically real analysis
. In that context, a function of a real variable was usually meant a real-valued function of a real variable, that is, a function whose domain and codomain
were the real numbers. However, because of their convenience in fields such as Fourier analysis, it was also common to consider complex functions of a real variable, that is, a function whose domain was the real numbers and whose range was the complex numbers.
Until the introduction of functional analysis
in the 1920s, the study of functions of a real variable was one of the two major subdivisions of mathematical analysis. The other was the study of functions of a complex variable, otherwise known as complex analysis
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a function of a real variable is a mathematical function whose domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
is the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
. More loosely, a function of a real variable is sometimes taken to mean any function whose domain is a subset of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
.
Functions of a real variable were the classical object of study in mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, specifically real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
. In that context, a function of a real variable was usually meant a real-valued function of a real variable, that is, a function whose domain and codomain
Codomain
In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
were the real numbers. However, because of their convenience in fields such as Fourier analysis, it was also common to consider complex functions of a real variable, that is, a function whose domain was the real numbers and whose range was the complex numbers.
Until the introduction of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
in the 1920s, the study of functions of a real variable was one of the two major subdivisions of mathematical analysis. The other was the study of functions of a complex variable, otherwise known as complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
.