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| page = 122
| publisher = John Wiley & Sons
| isbn = 978-1-119-58294-6 }}</ref>
In the context where it is defined, the [[derivative]] of a function is a measure of the rate of change of function values with respect to change in input values. Because a constant function does not change, its derivative is 0.<ref>{{cite book
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* 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.
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