Domain of a function

A well-defined function must map every element of its domain to an element of its codomain. For example, the function ${\displaystyle f}$ defined by

${\displaystyle f(x)={\frac {1}{x}}}$

has no value for ${\displaystyle f(0)}$. Thus, the set of all real numbers, ${\displaystyle \mathbb {R} }$, cannot be its domain. In cases like this, the function is either defined on ${\displaystyle \mathbb {R} \setminus \{0\}}$ or the "gap is plugged" by explicitly defining ${\displaystyle f(0)}$. If one extends the definition of ${\displaystyle f}$ to the piecewise function

${\displaystyle f(x)={\begin{cases}1/x&x\not =0\\0&x=0\end{cases}}}$

then f is defined for all real numbers, and its domain is ${\displaystyle \mathbb {R} }$.

Any function can be restricted to a subset of its domain. The restriction of ${\displaystyle g\colon A\to B}$ to ${\displaystyle S}$, where ${\displaystyle S\subseteq A}$, is written ${\displaystyle \left.g\right|_{S}\colon S\to B}$.

The natural domain of a function is the maximum set of values for which the function is defined, typically within the reals but sometimes among the integers or complex numbers. For instance the natural domain of square root is the non-negative reals when considered as a real number function. When considering a natural domain, the set of possible values of the function is typically called its range.[7]

Category theory deals with morphisms instead of functions. Morphisms are arrows from one object to another. The domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned or at least formulated more abstractly. For example, the notion of restricting a morphism to a subset of its domain must be modified. See subobject for more.

The word "domain" is used with other related meanings in some areas of mathematics. In real and complex analysis, a domain is an open connected subset of a real or complex vector space. In the study of partial differential equations, a domain is the open connected subset of the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ where a problem is posed, that is, where the unknown function(s) are defined.

As a partial function from the real numbers to the real numbers, the function ${\displaystyle x\mapsto {\sqrt {x}}}$ has domain ${\displaystyle x\geq 0}$. However, if one defines the square root of a negative number x as the complex number z with positive imaginary part such that z² = x, the function ${\displaystyle x\mapsto {\sqrt {x}}}$ has as its domain the entire real line (but now with a larger codomain). The domain of the trigonometric function ${\displaystyle \tan x={\frac {\sin x}{\cos x}}}$ is the set of all (real or complex) numbers not of the form ${\displaystyle {\frac {\pi }{2}}+k\pi ,k=0,\pm 1,\pm 2,\ldots }$.

• Bijection, injection and surjection
• Codomain
• Domain decomposition
• Attribute domain
• Effective domain
• Lipschitz domain
• Naive set theory

• Bourbaki, Nicolas (1970). Théorie des ensembles. Éléments de mathématique. Springer. ISBN 9783540340348.