PartialFunction 2002-08-23 20:48:14 2009-01-15 15:34:59 Definition total function domain of definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %\PMlinkescapeword{theory} A function $f:A\rightarrow B$ is sometimes called a \emph{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 \emph{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 \emph{domain of definition}.