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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.

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Hint : with this transformation you can take disk to upper half plane. And boundary of disk become real axis which is also real. Now by Schwartz reflection principle you can extend that to whole plane ..Just see what are assumption required to apply that principle, what are with us?

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