vector-valued function: a function whose domain is the set of real numbers and whose range is a set of vectors. The components of the vector-valued function are called component functions.
Often the independent variable will be denoted by t, since it will often be the case that we're dealing with time as the independent variable.
A space curve C is the plot of points (x,y,z), where
as t varies through an interval I. The equations for the coordinates are called parametric equations of C and t is called a parameter.
This curve can be considered the path of the tip of a vector-valued function
None to speak of.
Many of the ordinary rules of functions pass over directly to vector-valued functions: limits, continuity, etc.
Vector-valued functions are introduced, and some examples of space curves, which can be considered the paths of the tips of vector-valued functions, are given (e.g. twisted cubics, toroidal spirals, trefoil knots).
Many of the usual operations of real-valued functions pass directly over to vector-valued functions (e.g. continuity), only on a component-by-component basis.