$f$ holomorphic in $\mathbb{D}$ extends to Analytic function

Question

I have the following question:

Prove of give a counterexample: Suppose that $f$ is holomorphic on $\mathbb{D}$ and continuous on its closure. Then, $f$ extends to a analytic map on $B_{(0,R)}$ for some $R > 1$.

I am really not sure whether the statement is true. I can't think of any functions that are analytic in $\mathbb{D}$ but not on the boundary, so it seems like the claim might be true. But, I do not know how to define the function outside of $\mathbb{D}$.

Answer 1

You might try $f(z)=\sqrt{1-z}$.