I asked myself if analyticity on the upper half-line is needed in the analytic version of the Schwarz reflexion principle. Or can I only say that if $f:\{Im(z)\geq 0\}\rightarrow \Bbb{C}$ is continuous s.t. $f(\Bbb{R})\subset \Bbb{R}$ and on $\{Im(z)<0\}$ we define $$f(z)=\bar{f(\bar z)}$$ then $f$ is analytic on $\Bbb{C}$?
Somehow I think that this do not work so from the informations above I don't get analyticity on the par half plan which is needed in my opinion to solve this problem.
Or am I wrong?
Can maybe someone help me out here?
Isn't the counterexample you are looking for
$$f(z) = \bar{z}$$
This function is continuous on all of $\mathbb{C}$, moreover $f(z)=\bar{f(\bar{z})}$ on all $\mathbb{C}$. Likewise, $f(\mathbb{R})\subset\mathbb{R}$. However, $f(z)$ fails to be analytic on all of $\mathbb{C}$.