Total functions are those which are defined for all inputs in their domain. Normally, all functions are total because the domain of a function is the inputs for which it is defined.

So where does this distinction between total and partial functions come from?

In mathematics we rely on a human reader to understand the domain of a function and so we can be quite precise in our definition of a function’s domain. It’s easy enough to say that the function head returns the first element of any non-empty list.

However, the programming systems we use lack this level of precision when specifying types. They don’t allow us to specify that head takes a non-empty list whereas length takes any list at all. So, what happens is that head must be declared to have a domain of any list at all - empty or not - even though it isn’t defined for empty lists.

This kind of function - one whose domain is smaller than its type indicates - has been given the name “partial function” and thus the normal functions whose domain matches their type are called “total functions”.

However, this all assumes that the enforcement of the domain of the function is all up to the type checker of the programming system. In reality, the programmer himself is part of the system and must work to enforce the rules as well.

Thus the head function is total when you look at the bigger picture - one in which the precise domain of the function is enforced by the programmer, albeit without the usual assistance from the type checker. Just because the precise domain of the function is not visible to the type checker does not mean it is not known at all.

So, although these things we call “partial functions” aren’t just functions in principle, the term is still practical for referring to functions where the type checker can only restrict inputs to some superset of the true domain of the function. By calling out these functions we can better alert ourselves to the need for care in their use.

What about Total Functional Programming?

Total Functional Programming proposes the idea of “programming without partial functions”.

In practice, it just means that partial functions must become explicit, returning a Maybe or Option type in cases where the type checker cannot enforce the precise domain of the function. The benefit of this is that the programmer need not be so vigilant - they must handle the possibility of an undefined result. The downside is the code becomes more verbose in cases where the programmer has some certainty that the function IS defined for its input but is unable to communicate that to the type checker.