Minimum functions

Definition: A function of the form
$$z(x,y) = \min\{ax,by\},$$
where $a>0$ and $b>0$ is called a minimum function.

Remark: The function value is determined by the the element ($ax$ or $by$) with the smallest value. Therefore, we can rewrite the function as
$$z(x,y) = \left\{ \begin{array}{ll} ax & \text{if~}ax \leq by,\\ by & \text{if~}ax > by. \end{array} \right.$$

Example: The function $z(x,y) = \min\{3x, 4y\}$ is an example of a minimum function.