Example:Empty domain function

For any set [math]A [/math], there is exactly one function . It does not need a definition, because it is impossible to give it any input.

[math]f [/math] is vacuously well-defined: the definition is that for all , there is an value [math]f(x) \in A [/math]. This statement is vacuously true.

The uniqueness of [math]f [/math] is also vacuously true; to see this, expand the definition of uniqueness, and the definition of equality of functions.