I'm reading some lecture notes on complex analysis, where the following is stated:
An element of $\textrm{Aut}(\mathbb{H})$ is uniquely determined by the image of a point in $\mathbb{H}$ and of a point in $\partial \mathbb{H}$ (where the action of this group is extended by continuity).
As far as I know, $\mathrm{Aut}(\mathbb{H}) = \mathrm{PSL}(2, \mathbb{R})$, acting by homography, $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot z = \frac{az+b}{cz+d}.$$However, if I take $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} -\frac{3}{2} & -\frac{1}{4} \\ 1 & -\frac{1}{2}\end{pmatrix},$$then this automorphism isn't well-defined at $\frac{1}{2}$!
My question: Is there a way to recover this statement, or is it just wrong? For example, does adding a point at infinity help?
Hint: consider $\mathbb{H}\subset \mathbb{S}$ in the Riemann sphere. What is $\partial\mathbb{H}?$ In this situation, what is the image of $\frac12$ in your example?