I meet with term 'empty function' from time to time. It's high time to understand its nature. What is field( set of arguments) and what is image? ( set of value)?
The set theoretic definition of a function $f:A\rightarrow B$ is a special type of binary relation. I.e. it is a subset of $A \times B$. The empty set is also a subset of $A \times B$ and is defined as the empty function.