"Function" comes from mathematics, where it is a well-defined and total relation. The main place programmers use it wrong is that they often use the same term for partial functions, but that's easy to overlook, since a partial function is a 'kind' of function. (That and side effects, but those can be considered to be elided parts of the domain and codomain.)

What you are talking about might be called a relation, which is more general.