Functions and maps

Functions

In this course we define functions as objects which take an argument in $mathbf{R}^n$ , and return a value in . We use the notation

$f : mathbf{R}^n rightarrow mathbf{R},$

to refer to a function with ‘‘input’’ space $mathbf{R}^n$ . The ‘‘output’’ space for functions is .

Example: The function $f : mathbf{R}^2 rightarrow mathbf{R}$ with values

$f(x) = sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$

gives the distance from the point (x_1,x_2) to (y_1,y_2) .

We allow for functions to take infinity values. The domain of a function , denoted mbox{bf dom} f , is defined as the set of points where the function is finite.

Example:

Define the logarithm function as the function , with values if , and otherwise. The domain of the function is thus $mathbf{R}_{++}$ (the set of positive reals).

Maps

We reserve the term map to refer to functions which return more than a single value, and use the notation

$f : mathbf{R}^n rightarrow mathbf{R}^m,$

to refer to a map with input space $mathbf{R}^n$ and output space $mathbf{R}^m$ . The components of the map are the (scalar-valued) functions f_i , i=1,ldots,m .

Example: a map.