A partial function from a set to a set assigns at most one function output in to each function input in . It can be treated as an ordinary function on a specified subset , its domain of a partial function. The surrounding source set can contain points at which no value is defined.
For example, real division by the argument gives the partial function on with . Being undefined at zero is different from assigning a default value there. A partial function with is defined at every allowed input.
The union of a chain of partial-function graphs is a partial function: two values at the same input occur together in a larger chain member and therefore agree. Any closure or algebraic identity involving finitely many inputs passes to the union if it holds in every chain member, because one member contains all those inputs.
A weakly linear partial function on the real numbers has a domain closed under rational scalar multiplication and addition, and it preserves addition on that domain. A nonempty domain is a vector space over the rational numbers, and additivity implies rational homogeneity of an additive function by integer multiplication and division. The empty function also satisfies the defining conditions vacuously. This terminology does not assert homogeneity under arbitrary real scalars.
A partial unary operation on a set is a partial function with specified domain of a function and values in . For example, is a partial unary operation on the real numbers with .
A function preserving partial unary operations and must send each into and satisfy . It need not reflect definedness: a point outside can still map into . Preservation therefore requires both a domain of a partial function condition and a value equality.
For partial functions and , their function composition is defined precisely onFor example, composing with , as real partial functions, gives only on .
More generally, strict composition of partial functions makes defined only if every is defined and is defined on the resulting tuple. An argument discarded by the outer function must still be defined: the expression with the everywhere-zero constant function remains undefined at zero. A representation in a programming language using lazy evaluation therefore needs an explicit check of the intermediate definedness conditions.
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A **partial function** is a concept in mathematics and computer science that refers to a function that is not defined for all possible inputs from its domain. In other words, a partial function can provide an output for some inputs, but there are some inputs for which it does not produce an output at all. ### Key Characteristics of Partial Functions: 1. **Partial Domain**: The set of inputs for which the function is defined is known as its domain.