partial function

Noun

 * 1)  A function whose domain is a subset of the set on which it is formally defined; i.e., a function f: X→Y for which values f(x) are defined only for x ∈ W, where W ⊆ X.
 * 2) * 1967 [John Wiley & Sons],, Mathematical Logic, 2002, Dover, page 244,
 * The Church-Turing thesis applies to partial functions on the same grounds as to total functions (§ 41).
 * 1) * 1991, Michel Bidoit, Hans-Jörg Kreowski, Pierre Lescanne, Fernando Orejas, Donald Sannella (editors), Algebraic System Specification and Development: A Survey and Annotated Bibliography, Springer, LNCS 501, page 15,
 * Nowadays it seems quite clear that if algebraic specifications are to be used as a powerful and realistic tool for the development of complex systems they should permit the specification of partial functions.There are essentially two ways of specifying partial functions.
 * 1) * 2006, Paulo Oliva, Understanding and Using Spector's Bar Recursive Interpretation of Classical Analysis, Arnold Beckmann, Ulrich Berger, Benedikt Löwe, John V. Tucker (editors), Logical Approaches to Computational Barriers: 2nd Conference on Computability in Europe, Proceedings, Springer, LNCS 3988, page 432,
 * In the following $$\sigma$$ and $$\tau$$ will denote finite partial functions from $$\N$$ to $$\N$$, i.e. partial functions which are defined on a finite domain. A partial function which is everywhere undefined is denoted by $$\langle\ \rangle$$, whereas a partial function defined only at position $$k$$ (with value $$n$$) is denoted by $$\langle k,n\rangle$$.

Usage notes
This is not a formal term, but a metamathematical description which only assumes concrete meaning in context. For example, in computability theory, a partial function is a function whose domain is a subset of $${\N}^k$$ for some k, but in other fields the term has other meanings.

Translations

 * Icelandic: hlutskilgreint fall
 * Italian: funzione parziale