Constant function
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In mathematics a constant function is a function whose values do not vary and thus are constant. More formally, a function f : A → B, is a constant function if f(x) = f(y) for all x and y in A.
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.
For polynomial functions, a 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, it's derivative(s), 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.
References
- Herrlich, Horst and Strecker, George E., Category Theory, Allen and Bacon, Inc. Boston (1973)
- Template:Planetmath reference