Consider the

möbius function given by

**A** = [[

*a b*] [

*c d*]], all

4 parameters

real. If det

**A**=

0 then it is

constant.

Otherwise, it has a pole at *x=-d/c* (which you can treat as going to infinity, and even in a rigourous manner). Other than that, is it increasing or decreasing?

Take the derivative of the function to find out. It turns out that it's det **A**/(*cx+d*)^{2}. The denominator is always positive, so all that matters is the sign of the determinant of **A**! If it is positive the function is increasing; if it is negative, it is decreasing.

In any case it is one to one (assuming we deal with infinity and the pole in a proper fashion). Of course, since we can easily calculate the inverse function (it's also a möbius function, in case you hadn't guessed), this point is moot.