On Asymptotic Notation with Multiple Variables

We show that it is impossible to define big-O notation for functions on more than one variable in a way that implies the properties commonly used in algorithm analysis. We also demonstrate that common definitions do not imply these properties even if the functions within the big-O notation are restricted to being strictly nondecreasing. We then propose an alternative definition that does imply these properties whenever the function within the big-O notation is strictly nondecreasing.