Nullfunktion

Nullfunktion refers to a function that assigns the zero element of the codomain to every input. Formally, for a function f: X -> Y, f is a Nullfunktion if f(x) = 0_Y for all x in X, where 0_Y denotes the additive identity in Y.

Existence and uniqueness depend on the codomain having a zero element. If Y contains an additive identity, the Nullfunktion from X to Y exists and is unique for that pair (X, Y). It is a constant function, taking the same value for all inputs.

In algebraic contexts, the Nullfunktion plays a central role as the zero element in the set of all functions from X to Y, denoted by Hom(X, Y) when addition is defined pointwise. If Y is a vector space or an abelian group, the Nullfunktion is linear, because f(x) = 0_Y for all x implies f(x + x') = f(x) + f(x') and f(αx) = αf(x).

Examples include the real-valued case f: R -> R with f(x) = 0 for every x, or more generally any f: X -> Y with f(x) constant at the zero element of Y. Such a map is often called the zero map or zero transformation in linear algebra, where it serves as the additive identity in the space of all linear maps.

Notes: In some categorical settings, the notion of a zero morphism generalizes Nullfunktionen to categories that are pointed, where a distinguished zero object yields a well-defined zero morphism between any pair of objects.