Definition. Let f:AB and g:CD be partial functionsMathworldPlanetmath. g is said to be a subfunction of f if


In other words, g is a subfunction of f iff whenever xC such that g(x) is defined, then xA, f(x) is defined, and g(x)=f(x).

If we set C=AC and D=BD, then gf(C×D), so there is no harm in assuming that C and D are subsets of A and B respectively, which we will do for the rest of the discussion.

In practice, whenever g is a subfunction of f, we often assume that g and f have the same domain and codomainMathworldPlanetmath. Otherwise, we would specify that g is a subfunction of f:AB with domain C and codomain D.

For example, f: defined by


is a partial function, whose domain of definition is (-,-1][1,), and the partial function g: given by


is a subfunction of f. The domain of definition of g is (-,-1)(1,).

Two immediate properties of a subfunction g:CD of f:AB are

  • the range of g is a subset of the range of f:

  • the domain of definition of g is a subset of the domain of definition of f:


Definition. A subfunction g:CD of f:AB is called a restriction of f relative to D, if g(C)=f(C)D, and a restriction of f if g(C)=f(C).

Every partial function g:CD corresponds to a unique restriction g:Cg(C) of g.

A restriction g:CD of f:AB is certainly a restriction of f relative to D, since f(C)D=g(C)D=g(C), but not conversely. For example, let A be the set of all non-negative integers and -A:A2A the ordinary subtraction. -A is easily seen to be a partial function. Let B be the set of all positive integers. Then -B:B2B is a restriction of -A:A2A, relative to B. However, -B is not a restriction of -A, for n-Bn is not defined, while n-An=0A.


Title subfunction
Canonical name Subfunction
Date of creation 2013-03-22 18:41:54
Last modified on 2013-03-22 18:41:54
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 08A55
Classification msc 03E20
Defines restriction