There are two ways of approaching the concept of function. The ancient way is through formulae, while the modern approach is through the study of functions of sets of points. Most functions that we study arise naturally and can be defined formally (i. e. by formulae or expressions). On the other hand, many of the properties demanded of functions are best defined by thinking of them as set functions. Moreover, most of the formulae have a ``life of their own''; the formal expressions have a more general validity than as functions alone. Thus the study of formulae becomes algebra while the study of functions becomes analysis. Calculus is thus seen differently by algebraists and analysts. The fundamental example in both cases is that of a polynomial which we study below.

Kapil H. Paranjape 2001-01-20