subfunction
Definition. Let $f:A\to B$ and $g:C\to D$ be partial functions^{}. $g$ is said to be a subfunction of $f$ if
$$g\subseteq f\cap (C\times D).$$ 
In other words, $g$ is a subfunction of $f$ iff whenever $x\in C$ such that $g(x)$ is defined, then $x\in A$, $f(x)$ is defined, and $g(x)=f(x)$.
If we set ${C}^{\prime}=A\cap C$ and ${D}^{\prime}=B\cap D$, then $g\subseteq f\cap ({C}^{\prime}\times {D}^{\prime})$, 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 codomain^{}. Otherwise, we would specify that $g$ is a subfunction of $f:A\to B$ with domain $C$ and codomain $D$.
For example, $f:\mathbb{R}\to \mathbb{R}$ defined by
$$f(x)=\sqrt{{x}^{2}1}$$ 
is a partial function, whose domain of definition is $(\mathrm{\infty},1]\cup [1,\mathrm{\infty})$, and the partial function $g:\mathbb{R}\to \mathbb{R}$ given by
$$g(x)=\frac{{x}^{2}1}{\sqrt{{x}^{2}1}}$$ 
is a subfunction of $f$. The domain of definition of $g$ is $(\mathrm{\infty},1)\cup (1,\mathrm{\infty})$.
Two immediate properties of a subfunction $g:C\to D$ of $f:A\to B$ are

•
the range of $g$ is a subset of the range of $f$:
$$g(C)\subseteq f(C),$$ 
•
the domain of definition of $g$ is a subset of the domain of definition of $f$:
$${g}^{1}(D)\subseteq {f}^{1}(D).$$
Definition. A subfunction $g:C\to D$ of $f:A\to B$ is called a restriction of $f$ relative to $D$, if $g(C)=f(C)\cap D$, and a restriction of $f$ if $g(C)=f(C)$.
Every partial function $g:C\to D$ corresponds to a unique restriction ${g}^{\prime}:C\to g(C)$ of $g$.
A restriction $g:C\to D$ of $f:A\to B$ is certainly a restriction of $f$ relative to $D$, since $f(C)\cap D=g(C)\cap D=g(C)$, but not conversely. For example, let $A$ be the set of all nonnegative integers and ${}_{A}:{A}^{2}\to A$ the ordinary subtraction. ${}_{A}$ is easily seen to be a partial function. Let $B$ be the set of all positive integers. Then ${}_{B}:{B}^{2}\to B$ is a restriction of ${}_{A}:{A}^{2}\to A$, relative to $B$. However, ${}_{B}$ is not a restriction of ${}_{A}$, for $n{}_{B}n$ is not defined, while $n{}_{A}n=0\in A$.
References
 1 G. Grätzer: Universal Algebra^{}, 2nd Edition, Springer, New York (1978).
Title  subfunction 

Canonical name  Subfunction 
Date of creation  20130322 18:41:54 
Last modified on  20130322 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 