I am looking for an example of a function $f$ that is 1) continuous on the closed unit disk, 2) analytic in the interior and 3) cannot be extended analytically to any larger set. A concrete example would be the best but just a proof that some exist would also be nice. (In fact I am not sure they do.)


I know of examples of analytic functions that cannot be extended from the unit disk. Take a lacuanary power series for example with radius of convergence 1. But I am not sure if any of them define a continuous function on the closed unit disk. 



Example of continuous function that is analytic on the interior but cannot be analytically continued?
Let $f(z) = \sum z^n/n^2$, which is continuous and bounded on the closed unit disc but not analytic near $1$. Then consider 


$$\sum f(z^n)/n^2.$$


This should have a singularity at every root of unity; and should be analytic in the interior because it is uniformly convergent.


