Prove function with removable singularities is holomorphic

Question

I usually have trouble writing in detail why a specific function with removable singularities is holomorphic. Let $f$ be a holomorphic fuunction in the unit disk $\mathbb{D}$, where $f(\mathbb{D})\subset \mathbb{D}$ and $f$ has zeros at $a_1,\ldots,a_n\in \mathbb{D}$. Let $$B(z)=\prod_{j=1}^n\frac{z-a_j}{1-z\overline{a_j}}$$ and finally let $g(z)=\frac{f(z)}{B(z)}$. I want to prove that $g(z)$ is holomorphic in $\mathbb{D}$. I know that the singularities $a_1,\ldots, a_n$ are removable (proved it using Riemann's theorem) and I know about a corollary that says that if $g$ is holomorphic in a domain except for some removable singularities and is continuous in the whole domain, then it is holomorphic in the whole domain. However I am having trouble proving that it is continuous. Can someone help me? Is there another (maybe easier) way to prove that $g$ is holomorphic in $\mathbb{D}$?