Does there exists a function $f:\mathbb{\bar D}\to\mathbb{C}$ such that
(1) $f$ is holomorphic in $\mathbb{D}$ and $f$ is continuous in $\mathbb{\bar D}$ where $\mathbb{D}$ is the unit disk in the complex plane;
(2) for any $\delta>0$, $f$ has no analytic continuation to $B(0,1+\delta)$.
Since I cannot prove whether there is such a function or not, I tried some constructions. My first attempt was to construct a power series that has convergence radius $1$ and continuous in the closure of the unit disk. But my results so far have at least one singularity on the boundary. Then I tried to find an entire function with infinitely many zeros, and then identify its domain with the unit disk by a conformal equivalence, for example,
$$\sin\pi\frac{1-z}{1+z}$$
Then there is a convergence point of the zeros on the boundary. But the singularity problem still remains unsolved.
Any help is appreciated.
Many such functions are known. One is $\sum_{n=0}^{\infty} a_n \eta_n z^{n}$ where $a_n=1$ if $n$ is a power of $2$, $0$ otherwise and $\eta_n=e^{-\sqrt n}$. See 'natural boundary' in Rudin's RCA. There are simpler examples for your question but in this example the function cannot be extended analytically any open set containing the closed unit disk.