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

R^{n} to points in R

^{m}, the ith component function of f, denoted f

^{i}, is defined as follows:

For all points x in A, if f(x) = (a_{1}, ... , a_{m}), then f^{i}(x) = a_{i}.

To put it another way, f(x) = (f^{1}(x), ... , f^{m}(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), f^{1}(x,y) = 2x + y, which for specific values of x and y yields a number in R.

The jth partial derivative of the ith component function of f is denoted D_{j}f^{i}. This term is useful because it can be shown that if f is differentiable, each D_{j}f^{i} exists, and while the converse is not true, a slightly stronger condition is sufficient to guarantee that a function is continuously differentiable.