Let $G\subseteq \mathbb{H}$ be a domain, and suppose furthermore that $\delta G\bigcap \mathbb{R}\ne \mathrm{\varnothing }$, call this intersection $K$. Suppose $f:G\to \mathbb{C}$ is holomorphic, that $f(K)\subseteq \mathbb{R}$, and that it can be continuously extended to $f:cl(G)\to \mathbb{C}$ . 1;;Then we can extend $F:cl(G)\bigcup \bar{G}\to \mathbb{C}$ such that for $z\in \bar{G}$, we have that $F(z)\underset{def}{=}\overline{f(\stackrel{―}{z})}$.