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.