Let $f$ be analytic in $\mathbb{D}$ and continuous on $\mathbb{\overline{D}}$. Also, $f$ takes real value on $\partial\mathbb{D}$. Prove that $f$ can be extended to an entire function.
I tried to prove this with Cayley transformation. Since we know $\phi(z)=\dfrac{i-z}{i+z}$ maps the upper half plane $\mathbb{H}$ to $\mathbb{D}$, and $\mathbb{R}$ to $\partial\mathbb{D}$, consequently we know it's inverse $\phi^{-1}(z)=i\dfrac{1-z}{1+z}$ maps $\mathbb{D}\rightarrow\mathbb{H}$ and $\partial\mathbb{D}\rightarrow\mathbb{R}$.
I know this is a possibly duplicate with this and very closely related to this, but I'm having trouble understanding the whole process. I understand that I need to use the Schwarz reflection principle, which I can find from Wikipedia, but I'm not entirely sure on how to use it properly in this sense.
