EXAMPLES OF FUNCTIONS The examples given here: ¨ are basic types
of functions that are used everywhere in abstract math ¨ are
functions you need to be familiar with ¨ are (sigh) boring. For
any set A, the identity function is the function that takes an
element to itself; in other words, for every element , . Its
graph is the diagonal of . ¨ Warning: The word identity has two
other commonly used meanings. This causes trouble because people
may refer to the identity function as simply “the identity”,
especially in conversation. ¨ The notation for the identity
function on A is fairly common, but so is . ¨ The identity
function on a set A is the function that does nothing to each
element of A. ¨ The identity function on is the familiar
function defined by . Its graph in the plane is the diagonal
line from lower left to upper right through the origin. Its
derivative is the constant function defined by g(x) = 1. ¨ There
is a different identity function for each different set. See
overloaded notation. These function