In mathematics, the **domain** or **set of departure** of a function is the set into which all of the input of the function is constrained to fall.^{ [1] } It is the set X in the notation *f*: *X* → *Y*, and is alternatively denoted as .^{ [2] } Since a (total) function is defined on its entire domain, its domain coincides with its domain of definition.^{ [3] } However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain.

A domain is part of a function f if f is defined as a triple (*X*, *Y*, *G*), where X is called the *domain* of f, Y its * codomain *, and G its * graph *.^{ [4] }

A domain is not part of a function f if f is defined as just a graph.^{ [5] }^{ [6] } For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (*X*, *Y*, *G*). With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form *f*: *X* → *Y*.^{ [7] }

For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases).

If the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the *x*-axis.

A well-defined function must map every element of its domain to an element of its codomain. For example, the function defined by

has no value for . Thus the set of all real numbers, , cannot be its domain. In cases like this, the function is either defined on , or the "gap is plugged" by defining explicitly. For example. if one extends the definition of to the piecewise function

then * is defined for all real numbers, and its domain is .*

Any function can be restricted to a subset of its domain. The restriction of to , where , is written as .

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 and cannot directly connect to the domain outside *D*, including the point of the domain boundary , 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 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.

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 where a problem is posed (i.e., where the unknown function(s) are defined).

As a partial function from the real numbers to the real numbers, the function has domain . However, if one defines the square root of a negative number *x* as the complex number *z* with positive imaginary part such that *z*^{2} = *x*, then the function has the entire real line as its domain (but now with a larger codomain). The domain of the trigonometric function is the set of all (real or complex) numbers, that are not of the form .

- ↑ Codd, Edgar Frank (June 1970). "A Relational Model of Data for Large Shared Data Banks" (PDF).
*Communications of the ACM*.**13**(6): 377–387. doi:10.1145/362384.362685 . Retrieved 2020-04-29. - ↑ "Compendium of Mathematical Symbols".
*Math Vault*. 2020-03-01. Retrieved 2020-08-28. - ↑ Paley, Hiram; Weichsel, Paul M. (1966).
*A First Course in Abstract Algebra*. New York: Holt, Rinehart and Winston. p. 16. - ↑ Bourbaki 1970 , p. 76
- ↑ Bourbaki 1970 , p. 77
- ↑ Forster 2003 , pp. 10–11
- ↑ Eccles 1997 , p. 91 (quote 1, quote 2); Mac Lane 1998 , p. 8; Mac Lane, in Scott & Jech 1967 , p. 232; Sharma 2004 , p. 91; Stewart & Tall 1977 , p. 89
- ↑ Rosenbaum, Robert A.; Johnson, G. Philip (1984).
*Calculus: basic concepts and applications*. Cambridge University Press. p. 60. ISBN 0-521-25012-9. - ↑ Weisstein, Eric W. "Domain".
*mathworld.wolfram.com*. Retrieved 2020-08-28.

In mathematics, a **bijection**, **bijective function**, **one-to-one correspondence**, or **invertible function**, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function *f*: *X* → *Y* is a one-to-one (injective) and onto (surjective) mapping of a set *X* to a set *Y*. The term *one-to-one correspondence* must not be confused with *one-to-one function*.

**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 algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

In mathematics, a **partial function**f from a set X to a set Y is a function from a subset S of X to Y. The subset S, that is, the domain of f viewed as a function, is called the **domain of definition** of f. If S equals X, that is, if f is defined on every element in X, then f is said to be **total**.

In mathematics, **real analysis** is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In mathematics, a function *f* from a set *X* to a set *Y* is **surjective**, if for every element *y* in the codomain *Y* of *f*, there is at least one element *x* in the domain *X* of *f* such that *f*(*x*) = *y*. It is not required that *x* be unique; the function *f* may map one or more elements of *X* to the same element of *Y*.

In mathematics, an **injective function** is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of *at most* one element of its domain. The term *one-to-one function* must not be confused with *one-to-one correspondence* that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

In mathematics, the **codomain** or **set of destination** of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation *f*: *X* → *Y*. The term range is sometimes ambiguously used to refer to either the codomain or image of a function.

In mathematics, the **graph** of a function is the set of ordered pairs , where . In the common case where and are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset of this plane.

In mathematics, a **function** is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

In mathematics, more specifically topology, a **local homeomorphism** is a function between topological spaces that, intuitively, preserves local structure. If is a local homeomorphism, is said to be an **étale space** over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.

In mathematics, the **range of a function** may refer to either of two closely related concepts:

The theory of **functions of several complex variables** is the branch of mathematics dealing with complex-valued functions on the complex coordinate space of n-tuples of complex numbers.

In mathematics, a **map** is often used as a synonym for a function, but may also refer to some generalizations. Originally, this was an abbreviation of **mapping**, which often refers to the action of applying a function to the elements of its domain. This terminology is not completely fixed, as these terms are generally not formally defined, and can be considered to be jargon. These terms may have originated as a generalization of the process of making a geographical map, which consists of *mapping* the Earth surface to a sheet of paper.

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 ℝ, or a subset of 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.

In mathematics, the **restriction** of a function is a new function, denoted or , obtained by choosing a smaller domain *A* for the original function .

In commutative algebra, an element *b* of a commutative ring *B* is said to be **integral over***A*, a subring of *B*, if there are *n* ≥ 1 and *a*_{j} in *A* such that

In mathematics, the **hypograph** or **subgraph** of a function is the set of points lying on or below its graph. A related definition is that of such a function's epigraph, which is the set of points on or above the function's graph.

In mathematics, a **real-valued function** is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a **function of several real variables** or **real multivariate function** is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex valued functions may be easily reduced to the study of the real valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real valued functions will be considered in this article.

In mathematics, a **harmonic morphism** is a (smooth) map between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps i.e. those that are horizontally (weakly) conformal.

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

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