Editing Constant function (section)

== Other properties ==
For functions between [[preorder|preordered sets]], constant functions are both [[order-preserving]] and [[order-reversing]]; conversely, if {{math|''f''}} is both order-preserving and order-reversing, and if the [[Domain of a function|domain]] of {{math|''f''}} is a [[lattice (order)|lattice]], then {{math|''f''}} must be constant.

* Every constant function whose [[Domain of a function|domain]] and [[codomain]] are the same set {{math|''X''}} is a [[left zero]] of the [[full transformation monoid]] on {{math|''X''}}, which implies that it is also [[idempotent]].
* It has zero [[slope]] or [[gradient]]. 
* Every constant function between [[topological space]]s is [[continuous function (topology)|continuous]].
* A constant function factors through the [[singleton (mathematics)|one-point set]], the [[terminal object]] in the [[category of sets]]. This observation is instrumental for [[F. William Lawvere]]'s axiomatization of set theory, the [[Elementary Theory of the Category of Sets]] (ETCS).<ref>{{cite arXiv|last1=Leinster|first1=Tom|title=An informal introduction to topos theory|date=27 Jun 2011|eprint=1012.5647|class=math.CT}}</ref> 
* For any non-empty {{math|''X''}}, every set {{math|''Y''}} is [[isomorphic]] to the set of constant functions in <math>X \to Y</math>. For any {{math|''X''}} and each element {{math|''y''}} in {{math|''Y''}}, there is a unique function <math>\tilde{y}: X \to Y</math> such that <math>\tilde{y}(x) = y</math> for all <math>x \in X</math>. Conversely, if a function <math>f: X \to Y</math> satisfies <math>f(x) = f(x')</math> for all <math>x, x' \in X</math>, <math>f</math> is by definition a constant function.
** As a corollary, the one-point set is a [[generator (category theory)|generator]] in the category of sets.
** Every set <math>X</math> is canonically isomorphic to the function set <math>X^1</math>, or [[hom set]] <math>\operatorname{hom}(1,X)</math> in the category of sets, where 1 is the one-point set. Because of this, and the adjunction between Cartesian products and hom in the category of sets (so there is a canonical isomorphism between functions of two variables and functions of one variable valued in functions of another (single) variable, <math>\operatorname{hom}(X \times Y, Z) \cong \operatorname{hom}(X(\operatorname{hom}(Y, Z))</math>) the category of sets is a [[closed monoidal category]] with the [[Cartesian product]] of sets as tensor product and the one-point set as tensor unit. In the isomorphisms <math>\lambda: 1 \times X \cong X \cong X \times 1: \rho</math> [[natural transformation|natural in {{math|''X''}}]], the left and right unitors are the projections <math>p_1</math> and <math>p_2</math> the [[ordered pair]]s <math>(*, x)</math> and <math>(x, *)</math> respectively to the element <math>x</math>, where <math>*</math> is the unique [[point (mathematics)|point]] in the one-point set.

A function on a [[connected set]] is [[locally constant]] if and only if it is constant.
<!--Lfahlberg 01.2014: Perhaps needs information contained in: http://mathworld.wolfram.com/ConstantMap.html, http://www.proofwiki.org/wiki/Definition:Constant_Mapping, http://math.stackexchange.com/questions/133257/show-that-a-constant-mapping-between-metric-spaces-is-continuous  and programming http://www.w3schools.com/php/func_misc_constant.asp, http://www2.math.uu.se/research/telecom/software/stcounting.html -->