I'm studying Tom Apostol's "Modular Functions and Dirichlet Series in Number Theory", and there's one statement in Chapter 2 that I do not understand.
Let $R_\Gamma$ be the usual fundamental region for the modular group, and let $A = \{\tau \in R_\Gamma : \text{Re}(\tau) < 0\}$. Then Klein's modular function $J$ maps $A$ onto the upper-half plane $H$.
Apostol's proof involves showing that the three edges of $A$ are mapped onto $(-\infty, 0], [0, 1]$ and $[1, \infty)$ respectively, which I am fine with. Now comes the part I do not understand. He says that "As the boundary of $R_\Gamma$ is traversed counterclockwise the points inside $R_\Gamma$ lie on the left, hence the image points lie above the real axis in the image plane".
I am familiar with such arguments when applied using conformal maps, which are orientation-preserving, but $J$ is not conformal (since, for instance, it does not preserve the angles between the edges of the region $A$), so I don't see why the reasoning applies.
I have been able to cook up a more complicated proof by considering directly a point of the form $u + iv \in A$ for $-\frac{1}{2} < u < 0$, but I would like to understand Apostol's argument, as it seems much more elegant than my own.