We describe an axiomatic theory for the concept of one-place, partial function, where function is taken in its extensional sense. The theory is rather general; i.e., concepts such as natural number and set are definable, and topics such as non-strictness and self application can be handled. It contains a model of the (extensional) lambda calculus, and commonly applied mechanisms (such as currying and inductive definability) are possible. Furthermore, the theory is equi-consistent with and equally powerful as ZFC Set Theory. The theory (called (Axiomatic Function Theory, AFT) is described in the language of classical first order predicate logic with equality and one non-logical symbol for function application. By means of some notational conventions, we describe a method within this logic to handle undefinedness in a natural way.