In mathematics a constant function is a function whose values do not vary and thus are constant. For example, if f(x) = 4 for any x, then f is constant since any value in the domain is mapped to 4. More formally, a function f : A → B, is a constant function if f(x) = f(y) for all x and y in A. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

Notice that every empty function, that is, any function whose domain equals the empty set, is included in the above definition vacuously, since there are no x and y in A for which f(x) and f(y) are different. However some find it more convenient to define constant function so as to exclude empty functions. In the case where the domain of the function is the empty set {}, there is only one function with that domain (given any codomain), the empty function, and any formula can be used to define the empty function, since the formula wont apply to anything and will therefore never... Domain has several meanings: some kind of territory, such as (for example) a demesne or a realm In New Zealand a Town Domain is typically a public sport area administered by a Domain Board. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... Informally, a logical statement is vacuously true if it is true but doesnt say anything; examples are statements of the form everything with property A also has property B, where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly. ...

For polynomial functions, a non-zero constant function is called a polynomial of degree zero. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

Properties

Constant functions can be characterized with respect to function composition in two ways. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

The following are equivalent:

1. f : A → B, is a constant function.
2. For all functions g, h : C → A, f o g = f o h, (where "o" denotes function composition).
3. The composition of f with any other function is also a constant function.

The first characterization of constant functions given above, is taken as the motivating and defining property for the more general notion of constant morphism in Category theory. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics a constant function is a function whose values do not vary and thus are constant. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

In contexts where it is defined, the derivative of a function measures how that function varies with respect to the variation of some argument. It follows that, since a constant function does not vary, it's derivative(s), where defined, will be zero. Thus for example: In mathematics, the derivative is one of the two central concepts of calculus. ...

• If f is a real-valued function of a real variable, defined on some interval, then f is constant if and only if the derivative of f is everywhere zero.

For functions between preordered sets, constant functions are both order-preserving and order-reversing; conversely, if f is both order-preserving and order-reversing, and if the domain of f is a lattice, then f must be constant. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, the derivative is one of the two central concepts of calculus. ... This article is about the mathematics concept. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... Domain has several meanings: some kind of territory, such as (for example) a demesne or a realm In New Zealand a Town Domain is typically a public sport area administered by a Domain Board. ... The term lattice derives from the shape of the Hasse diagrams that result from depicting these orders. ...

Other properties of constant functions include:

• Every constant function whose domain and codomain are the same is idempotent.
• Every constant function between topological spaces is continuous.

Domain has several meanings: some kind of territory, such as (for example) a demesne or a realm In New Zealand a Town Domain is typically a public sport area administered by a Domain Board. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ...

References

• Herrlich, Horst and Strecker, George E., Category Theory, Allen and Bacon, Inc. Boston (1973)
• Constant function on PlanetMath.