[arketyp]

In general, couldn't a function be monotonically increasing in some interval and still have many points such that y = x? A curve that jiggles around the line y = x for instance.

[deredede]

We are not looking for points such that y = x but for distinct points x, x' that have the same y (f(x) = f(x'))

[chx]

So while the original problem was x^y=y^x he have rearranged this into ln x / x = ln y / y and so now are investigating the ln x / x function. Yes, the lettering could be clearer because y at some points means one variable and later on it means the value of the function.