Constancy on a closed subdisk of $\mathbb{D}$ implies constancy on $\mathbb{D}$

Show that if $f\in\Hol(\DD)$ is constant on a closed disk $r\bar\DD$ for some $0<r<1$, then $f$ is constant on $\DD$.

Solution

Let $c$ be the constant value of $f$ on $r\bar\DD$. Then the holomorphic function $f-c$ vanishes on the nonempty open set $r\DD$. Since $\DD$ is connected, the identity theorem gives $f-c\equiv0$ on $\DD$. Hence $f\equiv c$ on $\DD$.