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[edit] Mathematical definition of a function
The informal idea of a function as a rule has been used since ancient times and is still used as the definition of a function in informal contexts such as introductory calculus textbooks. A typical example of this informal definition is given by Tomas and Finney [1996]:
- "A function from a set D to a set R is a rule that assigns a unique element f(x) in R to each element x in D." (p. 18, emphasis from the original).
This informal definition is sufficient for many purposes, but it relies on the undefined concept of a "rule". In the late 1800s, the question of what constitutes a valid rule defining a function came to the forefront. The consensus of modern mathematicians is that the word "rule" should be interpreted in the most general sense possible: as an arbitrary binary relation.
Thus it is common in advanced mathematics (see Bartle [1976] for an example) to formally define a function f from a set D to a set C to be a set Gf of ordered pairs (x,y) in the Cartesian product 
such that for each x in D there is at most one pair (x,y) in the set Gf. The "rule" of the function is: given x in D, if there is a pair (x,y) in Gf then f(x) = y, and otherwise f(x) is not defined. The set Gf is called the graph of f. The set of x for which f is defined is called the domain of f; if the domain of f is all of D then f is called total and the notation Failed to parse (Can't write to or create math output directory): f\colon D \rightarrow C
is used. The set C is called the codomain of the function; this must be specified because it is not determined by Gf. The set of elements of C that occur as values of f is called the range of f.
Variations of this formal definition are sometimes more convenient for specific disciplines. In some contexts of category theory, even if a function f from a set D is not defined for every element of D (in which case f is a partial function on D), the set D is still called the domain of f. In set theory, it is common to identify the function f with its graph Gf; this identification removes the need to specify either D or C in the formal definition.
[edit] References
- Thomas, G. and Finney, R. Calculus and Analytic Geometry. Addison-Wesley, 1996.
- Bartle, R. The elements of real analysis. Wiley and Sons, 1976.
[edit] Harvard referencing
Soare (1951b) says
Soare, Robert I. (1951a), All the Presidents' Names, Home Base, New York: The Pentagon.
Soare, Robert I. (1951b), All the Presidents' Names, Home Base, New York: The Pentagon.
