  Partiality is a natural phenomenon in computability that we cannot get
around. So, the question is whether we can give the areas where partiality
occurs, that is, where non-termination happens, more structure. In this paper
we consider function classes which besides the total functions only contain
finite functions whose domain of definition is an initial segment of the
natural numbers. Such functions appear naturally in computation. We show that a
rich computability theory can be developed for these functions classes which
embraces the central results of classical computability theory, in which all
partial (computable) functions are considered. To do so, the concept of a
G\"odel number is generalised, resulting in a broader class of numberings. The
central algorithmic idea in this approach is to search in enumerated lists. In
this way, function computability is reduced to set listability. Besides the
development of a computability theory for the functions classes, the new
numberings -- called quasi-G\"odel numberings -- are studied from a
numbering-theoretic perspective: they are complete, and each of the function
classes numbered in this way is a retract of the G\"odel numbered set of all
partial computable functions. Moreover, the Rogers semi-lattice of all
computable numberings of the considered function classes is studied and results
as in the case of the computable numberings of the partial computable functions
are obtained. The function classes are shown to be effectively given algebraic
domains in the sense of Scott-Ershov. The quasi-G\"odel numberings are exactly
the admissible numberings of the computable elements of the domain. Moreover,
the domain can be computably mapped onto every other effectively given one so
that every admissible numbering of the computable domain elements is generated
by a quasi-G\"odel numbering via this mapping.
