Partial function

A partial function assigns each element of a specified subset of its inputs exactly one output. It is undefined for inputs outside that subset.

Function: A relation that assigns each input in its domain exactly one output in its codomain. A partial function is a function whose domain covers only some specified possible inputs.
Graph of a function: The set of ordered input-output pairs belonging to a function. A partial function’s graph contains pairs only for inputs in its domain.
Division by zero: An attempted division whose denominator is zero, for which ordinary real division is undefined. Real division is naturally partial because zero denominators have no assigned quotient.
Total function: A function defined for every element of its specified input set. It differs precisely by requiring the domain to include every possible input.
Domain of a function: The set of inputs for which a function is defined. The partial function’s domain is the subset on which it has values.
Function composition: The operation that applies one function to an output of another. Composition is defined only when the first output lies in the next function’s domain.
Square root: A value whose square equals a given number; over the reals, nonnegative inputs have nonnegative square roots. The real-valued square-root operation is undefined on negative inputs.
Multivalued function: A relation that may associate one input with several possible outputs. Partiality allows missing outputs, while multivaluedness allows more than one.
Codomain: The specified set in which a function’s outputs are required to lie. Even where the function is defined, its outputs must belong to its codomain.
Identity function: The function that maps every element of a set to itself. On a partial function’s domain, the identity gives a simple example with restricted availability.

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