In basic (that is to say, one-dimensional) calculus, a function
is a rule that assigns to every real number in its domain some
other real number. It is under this framework that certain
essential terms (continuous and differentiable functions, for
example) are defined. When moving from one-variable calculus to
multivariable calculus, we wish to preserve as many of these
concepts as possible. Unfortunately, many of these concepts are
defined in a way that assumes implicitly that the output of a
function will be a single real number. In order to generalize
these concepts, we can introduce the concept of a component
function. If f is a function that maps a set of points A in Rn
to points in Rm, the ith component function of f, denoted fi, is
defined as follows:  For all points x in A, if f(x) = (a1, ... ,
am), then fi(x) = ai.  To put it another way, f(x) = (f1(x), ...
, fm(x)) for all points x in A. Therefore, the range of these
component functions lies in R.  Example: If f(x,y) = (2x + y,
3xy), f1(x,y) = 2x +