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.

In the context of polynomial functions, a non-zero constant function is called a polynomial of degree zero.

A function is said to be identically zero if it takes the value 0 for every argument; it is then trivially a constant function.

Properties

Constant functions can be characterized with respect to function composition in two ways.

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 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 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.

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, PlanetMath.org.