Composition of functions on Riemann surfaces

Question

Let $X$ be a Riemann surface, $p\in X$, $U$ a neighbourood of $p$, $f:U\setminus\{p\}\to \mathbb{C}$ be a olomorphic function on the punctured set. How can I prove that $f$ has a removable singularity/pole/essential singularity in $p$ if and only if, for every chart $\phi:W\to V \subset \mathbb{C}$, with $p \in W$, $f \circ \phi^{-1}$ has a removable singularity/pole/essential singularity? Thanks