that it is also idempotent. It has zero slope/gradient. Every constant function between topological spaces is continuous. A constant function factors through the 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). For any non-empty Y, every set X is isomorphic to the set of constant functions in . For any Y and each element x in X, there is a unique function such that for all . Conversely, if a function satisfies for all , is by definition a constant function. As a corollary, the one-point set is a generator in the category of sets. Every set is canonically isomorphic to the function set , or hom set 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, ) 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 natural in X, the left and right unitors are the projections and the ordered pairs and respectively to the element , where is the unique point in the one-point set. A function on a connected set is locally constant if and only if it is constant. References Herrlich, Horst and Strecker, George E., Category Theory, Heldermann Verlag (2007). External links Elementary mathematics Elementary special functions Polynomial functions