nontotal

Nontotal is an adjective used in mathematics, logic, and computer science to describe a relation or function that is not defined for every element of its input domain. A function is total if every input yields a value in the codomain; if there exists at least one input for which no output is assigned, the function is nontotal (or, more commonly in some contexts, partial). The term highlights the property of non-definition rather than the specific mechanism by which the undefined inputs arise.

In formal discussions, the standard notion is that of a partial function. A partial function from a set A to a set B has a domain of definition D that is a subset of A; outside D, the function has no value. The label nontotal is sometimes used synonymously with partial, but some authors prefer partial to emphasize the mathematical object and reserve nontotal for broader or less formal contexts, such as a process that may fail to yield a result on certain inputs.

Examples are common in programming and logic. A function f: N -> N defined by f(n) = 1 if n is even and undefined if n is odd is nontotal. Similarly, a mathematical rule like “take the square root of n if n is a perfect square; otherwise undefined” is nontotal. In computability, non-termination is a typical source of non-totality, since a program may run forever for some inputs, failing to produce an output.

Handling non-totality often involves explicit error signaling, option or Maybe types, or restricting the input domain to guarantee totality. See also partial function, total function, domain of definition, halting problem.