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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]].
* 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>
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