Function of a real variable information

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers $\mathbb {R}$ , or a subset of $\mathbb {R}$ that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. 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 $\mathbb {R}$ -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 $\mathbb {R}$ -algebra, such as the complex numbers or the quaternions. The structure $\mathbb {R}$ -vector space of the codomain induces a structure of $\mathbb {R}$ -vector space on the functions. If the codomain has a structure of $\mathbb {R}$ -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.

Function of a real variable information

and 24 Related for: Function of a real variable information

Function of a real variable

Function of several real variables

Convex function

Real analysis

Real variable

Complex analysis

Exponential function

Homogeneous function

Differentiable function

Total variation

Random variable

Graph of a function

Holomorphic function

Inverse function

Piecewise linear function

Derivative

Quadratic growth

Shift operator

Laplace transform

Analytic function

Implicit function theorem

Domain of a function

Slowly varying function

Binary function

Function
$x \mapsto f (x)$
History of the function concept
Examples of domains and codomains
$X$ → $\mathbb {B}$ , $\mathbb {B}$ → $X$ , $\mathbb {B} ^{n}$ → $X$ $X$ → $\mathbb {Z}$ , $\mathbb {Z}$ → $X$ $X$ → $\mathbb {R}$ , $\mathbb {R}$ → $X$ , $\mathbb {R} ^{n}$ → $X$ $X$ → $\mathbb {C}$ , $\mathbb {C}$ → $X$ , $\mathbb {C} ^{n}$ → $X$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Binary relation Partial Multivalued Implicit Space
v t e