Analytic continuation of function continuous on boundary

Question

Suppose one has a function $f$ in the disc algebra ie: $f$ is continuous on $|z|\leq1$ and holomorphic in $|z|<1$. I wondered, can $f$ always be extended to a holomorphic function on some region containing the boundary (ie: $T$ the unit circle). If not can you give me an example? Thank you,

Answer 1

No, otherwise there would be no need to define the disc algebra. Consider the map $z\to -(1+z)/(1-z), $ which maps $\bar {\mathbb {D}}\setminus \{1\}$ onto the closed left half plane. The function $f(z) = \exp {[-(1+z)/(1-z)}]$ is then bounded and holomorphic in $\mathbb {D}$ and continuous on $\bar {\mathbb {D}}\setminus \{1\}.$ Note the abberant behavior of $f(z)$ as $z\to 1.$ We kill this off by defining $g(z) = (z-1)f(z).$ This $g$ is in the disc algebra. Check that the the rate of vanishing of $g$ along the radius terminating at $1$ is way too fast for any function holomorphic on a larger open disc (except for the zero function, which this isn't).