If $f$ is a continuous analytic function on a unit disc about the origin $D\setminus\{0\}$, then $f$ is also analytic at $0$.

Question

The question defined as $f: D \rightarrow \mathbb C$ is continuous function analytic on $D \setminus\{ 0\}$, where $D$ is a unit disc about the origin. How can I show that $f$ is also analytic at zero?

I searched the old topics but most of the time the function $f$ was constant or maps itself. I couldn't figure out what $f$ maps. Can we talk about a constant function in this case? Any help is greatly appreciated.