Let $f,g$ be holomorphic function in $mathbb{D}$ that are continuous in $overline{mathbb{D}}$. Show that if $f=g$ on $|z|=1$, then $f=g$

Let $f,g$ be holomorphic function in $mathbb{D}$ that are continuous in $overline{mathbb{D}}$. Show that if $f=g$ on $|z|=1$, then $f=g$

It seems like identity theorem. But they have to be equal on an open connected set. $overline{mathbb{D}}$ is not open so I cannot use the identity theorem. At least directly.