The following extract is from the proof of the classification theorem of the isometries of the hyperbolic plane from the book Fuchsian groups by Svetlana Katok (pages 8 to 10).
For an isometry $f$ we found that $$f(z) = z \qquad \text{or} \qquad f(z) = -\bar{z}$$ for $z \in \mathbb{H} := \{z \in \mathbb{C} : \mathrm{Im}(z) > 0\}$. Now the author concludes that only one equation can hold for all $z \in \mathbb{H}$ by the continuity of an isometry.
I do not quite see this. First of all, I think continuity is meant with respect to the hyperbolic metric $\rho$ (i.e. the one given by the Riemannian metric). Then continuity is obvious since each isometry is Lipschitz continuous. However, I do not see why this should imply that only one equation holds.
Edit. Thanks to reuns I figured out the main issue: I have to show that a function which is continuous with respect to the hyperbolic metric is also continuous with respect to the standard metric. However this is clear since the metric topology generated by the hyperbolic distance function is the same as the original topology on $\mathbb{H}$.