Singularities at infinity

Let \(R\ge0\) and let \(f\) be holomorphic on \(\{z\in\CC:\abs{z}>R\}\). Put \(g(w)\coloneqq f(1/w)\), a holomorphic function on the punctured disc \(\{w:0<\abs{w}<1/R\}\) (on \(\CC\setminus\{0\}\) if \(R=0\)). Then \(f\) has a removable singularity at \(\infty\), a pole of order \(n\) at \(\infty\), or an essential singularity at \(\infty\) if \(g\) has, respectively, a removable singularity, a pole of order $n$, or an essential singularity at \(w=0\).