If $f : \mathbb{C} \to \mathbb{C}$ is represented by a power series in $D$ (the unit disc), and $f$ extends continuously into $\bar D$, does the same power series represent $f$ in $\bar D$?
I suspect that the answer is no, since this would imply that $f$ then extends holomorphically into $\bar D$. However, this is not that great of a reason, and I can't think that of a specific counter example. (This question arose while trying to solve a different analysis exercise.)
I should also add that, after looking at the power series expansion theorem again, I was reminded that one only has a power series expansion for f in an open disc whose closure is contained in the region where f is holomorphic (otherwise the Cauchy integral formula doesn't apply). However, I'm not sure if this means that one cannot find a single power series expansion for $f$ in the whole disc, even if $f$ is not holomorphic on the boundary. This is possibly a separate question.
Duplicate here: Continuity of analytic function implies convergence of power series?
