TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the domain of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
is the set of inputs accepted by the function. It is sometimes denoted by $\operatorname\left(f\right)$, where is the function. More precisely, given a function $f\colon X\to Y$, the domain of is . Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that and are both subsets of $\R$, the function can be graphed in the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...
. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis. For a function $f\colon X\to Y$, the set is called the
codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, and the set of values attained by the function (which is a subset of ) is called its
range Range may refer to: Geography * Range (geographic)A range, in geography, is a chain of hill A hill is a landform A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...
or
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment Environment most often refers to: __NOTOC__ * Natural environment, all living and non- ...
. Any function can be restricted to a subset of its domain. The restriction of $f \colon X \to Y$ to $A$, where $A\subseteq X$, is written as $\left. f \_A \colon A \to Y$.

# Natural domain

If a
real function In mathematical analysis, and applications in geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. ...
is given by a formula, it may be not defined for some values of the variable. In this case, it is a
partial function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, and the set of real numbers on which the formula can be evaluated to a real number is called the natural domain or domain of definition of . In many contexts, a partial function is called simply a ''function'', and its natural domain is called simply its ''domain''.

## Examples

* The function $f$ defined by $f\left(x\right)=\frac$ cannot be evaluated at 0. Therefore the natural domain of is $\mathbb \setminus \$. * In contrast, if $f$ is the
piecewise In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
function $f\left(x\right) = \begin 1/x&x\not=0\\ 0&x=0 \end,$ then ''$f$'' is defined for all real numbers, and its natural domain is $\mathbb$. * The function $x\mapsto\sqrt x$ has as its natural domain the non-negative real numbers, which can be denoted by $\mathbb R_$, by the interval $\left(0,\infty\right)$, or by $\$. * The
tangent function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
$\tan x$ has as its natural domain the set of all real numbers which are not of the form $\tfrac + k \pi,$ where is any
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
.

# Other uses

The word "domain" is used with other related meanings in some areas of mathematics. In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, a domain is a connected
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. In
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

and
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
, a domain is an
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
connected subset of a
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

vector space. In the study of
partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s, a domain is the open connected subset of the
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
$\mathbb^$ where a problem is posed (i.e., where the unknown function(s) are defined).

# Set theoretical notions

For example, it is sometimes convenient in
set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...
to permit the domain of a function to be a
proper class Proper may refer to: Mathematics * Proper map In mathematics, a function (mathematics), function between topological spaces is called proper if inverse images of compact space, compact subsets are compact. In algebraic geometry, the analogous ...
, in which case there is formally no such thing as a triple . With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form ., p. 91 (
quote 1 Quote is a hypernym of quotation, as the repetition or copy of a prior statement or thought. Quotation marks are punctuation marks that indicate a quotation. Both ''quotation'' and ''quotation marks'' are sometimes abbreviated as "quote(s)". Co ...
quote 2 Quote is a hypernym In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of ling ...
; ,
p. 8P. is an abbreviation or acronym that may refer to: * Page (paper) A page is one side of a leaf A leaf (plural leaves) is the principal lateral appendage of the vascular plant plant stem, stem, usually borne above ground and specialized f ...
Mac Lane, in ,
p. 232P. is an abbreviation or acronym that may refer to: * Page (paper) A page is one side of a leaf A leaf (plural leaves) is the principal lateral appendage of the vascular plant plant stem, stem, usually borne above ground and specialized f ...
, p. 91 , p. 89/ref>

* Attribute domain *
Bijection, injection and surjection In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
*
Codomain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

*
Domain decomposition In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solutio ...
* Effective domain *
Image (mathematics) In mathematics, the image of a Function (mathematics), function is the set of all output values it may produce. More generally, evaluating a given function f at each Element (mathematics), element of a given subset A of its Domain of a functio ...
*
Lipschitz domainIn mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a Domain (mathematical analysis), domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lips ...
*
Naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sens ...
*
Support (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

# References

* {{Mathematical logic Functions and mappings
Basic concepts in set theory{{Commons This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed. Mathematical concepts ...