Domain of a function

Examples

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 defining ${\displaystyle f(0)}$ explicitly. For example. 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 ${\displaystyle f}$ 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 as ${\displaystyle \left.g\right|_{S}\colon S\to B}$.

Natural domain

The natural domain of a function (sometimes shortened as domain) is the maximum set of values for which the function is defined, typically within the reals but sometimes among the integers or complex numbers as well. 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.^[8] Also, in complex analysis especially several complex variables, when a function f is holomorpic on the domain ${\displaystyle D\subset \mathbb {C} ^{n}}$ and cannot directly connect to the domain outside D, including the point of the domain boundary ${\displaystyle \partial D}$, in other words, such a domain D is a natural domain in the sense of analytic continuation, the domain D is called the domain of holomorphy of f and the boundary is called the natural boundary of f.

Category theory

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. For more, see subobject.

Other uses

The word "domain" is used with other related meanings in some areas of mathematics. In topology, a domain is a connected open set.^[9] 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 (i.e., where the unknown function(s) are defined).

More common examples

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, then the function ${\displaystyle x\mapsto {\sqrt {x}}}$ has the entire real line as its domain (but now with a larger codomain). The domain of the trigonometric function ${\displaystyle \tan x={\tfrac {\sin x}{\cos x}}}$ is the set of all (real or complex) numbers, that are not of the form ${\displaystyle {\tfrac {\pi }{2}}+k\pi ,k=0,\pm 1,\pm 2,\ldots }$.

See also

• Attribute domain
• Bijection, injection and surjection
• Codomain
• Domain decomposition
• Effective domain
• Image (mathematics)
• Lipschitz domain
• Naive set theory
• Support (mathematics)

Notes

References

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