 Constant function

Not to be confused with Function constant.
In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f(x) = 4 is constant since f maps any value 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.
Every empty function is constant, vacuously, since there are no x and y in A for which f(x) and f(y) are different when A is the empty set. Some find it more convenient, however, to define constant function so as to exclude empty functions.
In the context of polynomial functions, a nonzero constant function is called a polynomial of degree zero.
Properties
Constant functions can be characterized with respect to function composition in two ways.
The following are equivalent:
 f : A → B is a constant function.
 For all functions g, h : C → A, f o g = f o h, (where "o" denotes function composition).
 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 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, its derivative, where defined, will be zero. Thus for example:
 If f is a realvalued 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 orderpreserving and orderreversing; conversely, if f is both orderpreserving and orderreversing, and if the domain of f is a lattice, then f must be constant.
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.
A function on a connected set is locally constant if and only if it is constant.
References
 Herrlich, Horst and Strecker, George E., Category Theory, Allen and Bacon, Inc. Boston (1973)
 Constant function on PlanetMath
By name Zero polynomial · Constant function · Linear equation · Quadratic function · Cubic function · Quartic function · Quintic function · Sextic equation · Septic equation · Octic equation · Nonic equation · Decic equation · Hectic equationBy degree Categories: Polynomials
 Functions and mappings
 Elementary mathematics
 Types of functions
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