Constant function: Difference between revisions

Revision as of 01:58, 27 April 2007

Revision as of 01:58, 27 April 2007

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In [[mathematics]] a '''constant function''' is a [[function (mathematics)|function]] whose values do not vary and thus are [[constant]]. For example, if we have the function ''f''(''x'') = 4, then ''f'' 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''.

Notice that every [[empty function]], that is, any function whose [[Domain (mathematics)|___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|polynomial function]]s, a non-zero 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, its derivative, 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.

For functions between [[preorder|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 (mathematics)|___domain]] of ''f'' is a [[lattice (order)|lattice]], then ''f'' must be constant.

Other properties of constant functions include:

* Every constant function whose [[Domain (mathematics)|___domain]] and [[codomain]] are the same is [[idempotent]].

* Every constant function between [[topological space]]s is [[continuous function (topology)|continuous]].

In a [[connected set]], a function 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)

*{{planetmath reference|id=4727|title=Constant function}}

[[Category:Functions and mappings]]

[[Category:Elementary mathematics]]

[[cs:Konstantní funkce]]

[[da:Konstant funktion]]

[[fr:Fonction constante]]

[[ko:상수 함수]]

[[it:Funzione costante]]

[[he:פונקציה קבועה]]

[[pl:Funkcja stała]]

[[sk:Konštantná funkcia]]

[[sr:Константна функција]]

[[zh:常数函数]]