You can use the proof of the Schwarz reflection principle to show that if $D$ is a domain in the complex plane, where $D^+$ denotes the subset of D above the real line, and $D^-$ the subset below, if $f:D\to C$ which is analytic on $D^+$ and $D^-$, and continuous on $D$, then $f$ is analytic on $D$.
I want to generalize this as follows: Let $D$ be a domain in the complex plane, and $D'$ an open dense subset of $D$. If $f$ is a function analytic on $D'$, and continuous on $D$, then $f$ is analytic on $D$.
The proof of the Schwarz reflection principle uses integration along arbitrary triangles, and since the intersection of a triangle with the real line is really simple, you can "cut the real line out" so to speak when you are taking the integral. But if all you know is that $f$ is analytic on an open dense set what can you do?