Is there an entire function $f$ such that $f(z)=1/z$ for all z in the plane punctured at $0$?
I am viewing it as a problem to analytically extend the function $1/z$ in the entire plane which is not allowed by Riemann's removable singularity theorem as $z=0$ is a pole here.
Am I thinking right? What am I missing? What are the ways to counter these type of questions?