Constant function: Difference between revisions

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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''.

#redirect [[Constant]

Notice that every [[empty function]], that is, any function whose [[___domain]] equals the [[empty set]], is included in the above definition [[vacuous truth|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''&nbsp;:&nbsp;''C'' &rarr; ''A'', ''f'' <small> o </small> ''g'' = ''f'' <small> o </small> ''h'', (where "<small>o</small>" 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 number|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.

Other properties of constant functions include:

* Every constant function is [[idempotent]].

* Every constant function between [[topological space]]s is [[continuous]].

==References==

Herrlich, Horst and Strecker, George E., ''Category Theory'', Allen and Bacon, Inc. Boston (1973)