Q:

The disk algebra and continuous functions

The disk algebra is the set of continuous functions $f: D \to \mathbb C$ where $D$ is the closed unit disc in $\mathbb C$ and $f$ is analytic on the interior of $D$. It is endowed with the $\sup$-norm. 
Why this extra restriction to only include $f$ that are analytic on the interior of $D$? Wouldn't $C(D)$, the set of all continuous $f:D\to \mathbb C$ make a fine complete normed algebra, too?

A:

$C(D)$ is a perfectly fine Banach algebra. (But $C(\text{open unit disc})$ is not, at least under the sup-norm.)

