Extension (or "counterexample") of the Schwarz reflection principle

Question

The usual statement for Schwarz reflection principle on a general analytic arc is something along these lines:

Given a holomorphic function $f$ on a region $\Omega$, assume that

1. The function has a continuous extention to an analytic arc $\gamma$ on $\partial \Omega$
2. $f(\gamma)$ is an analytic arc
3. $f(\Omega)$ is on one side of $f(\gamma)$

Then $f$ has an analytic extension across $\gamma$ (except at most its endpoints).

While the first two assumptions are clear to me, I do not understand the third one: it is not a requirement that we can trace back to the "original" Schwarz reflection principle.

Is there some deep reason, perhaps geometric, for this addition, or is it not strictly necessary, and thus we have some generalizations of the principle which weaken this assumption?