# Schwarz reflection on unit disk and Cayley transformation.

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.