Special case of the Monodromy Theorem

Question

Let $D$ be the unit disc in the complex plane. Suppose the following theorem is true: If $f \in H(D)$ and \begin{equation*} f(z)=\sum_{n=0}^\infty a_nz^n \end{equation*} has radius of convergence $1$, then $f$ has at least one singular point on the unit circle. How can we apply this result to prove this version of the Monodromy Theorem: If $\Omega$ and $D$ are concentric discs, $(f,D)$ is a function element which can be analytically continued along every curve in $\Omega$ that starts at the origin $0$, then there exists a $g \in H(\Omega)$ such that $g(z)=f(z)$ on $D$?