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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]] functions, a non-zero constant function is called a polynomial of degree zero.
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*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]].
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