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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 of a function|___domain]] of ''f'' is a [[lattice (order)|lattice]], then ''f'' must be constant.
* Every constant function whose [[Domain of a function|___domain]] and [[codomain]] are the same set ''X'' is a [[left zero]] of the [[full transformation monoid]] on ''X'', which implies that it is also [[idempotent]].
* It has no slope/[[Slope|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>
* Every set ''X'' is [[isomorphic]] to the set of constant functions into it. For each element ''x'' and any set ''Y'', there is a unique function <math>\tilde{x}: Y \to X</math> such that <math>\tilde{x}(y) = x</math> for all <math>y \in Y</math>. Conversely, if a function <math>f: Y \to X</math> satisfies <math>f(y) = f\left(y'\right)</math> for all <math>y, y' \in Y</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 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.
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