partial function

A function $f:A\to B$ is sometimes called a total function, to signify that $f(a)$ is defined for every $a\in A$. If $C$ is any set such that $C\supseteq A$ then $f$ is also a partial function from $C$ to $B$.

Clearly if $f$ is a function from $A$ to $B$ then it is a partial function from $A$ to $B$, but a partial function need not be defined for every element of its domain. The set of elements of $A$ for which $f$ is defined is sometimes called the domain of definition.