Let $\displaystyle f: D(P,r)$\$\displaystyle \{P\} \rightarrow C$ be holomorphic. Let $\displaystyle U=f(D(P,r)$\$\displaystyle \{P\})$. Assume that $\displaystyle U$ is open. Let $\displaystyle g:U \rightarrow C$ be holomorphic. If $\displaystyle f$ has a removable singularity at $\displaystyle P$, does $\displaystyle p \circ f$ have one also? what about the case of poles and essential singularities?