analytical function question

[eugene]

It is going to be a silly question, but i'm a bit unsure:

If we know that the function f(z) is analytical in 
D = {z: |z|<1} is it true that it is analytical in the closure of D^ = {z: 
|z|<=1} ?

As far as i understand if f iz analytical at every point z from D, then it is 
analytical in some neighborhood of z, and these neighborhoods cover the unit 
circle D^.
Where am i wrong?


Thanks