I am studying complex analysis and I am confused about why $f(z)=1/(z-z_0)$ is not analytic at $z=z_0$. For a function to be analytic, it must be differentiable and single-valued. Obviously, $f'(z)=-1/(z-z_0)^2$ is not defined at $z=z_0$, so the derivative does not exist. Also, $f(z)=1/(z-z_0)$ is not defined at $z=z_0$, so $f$ is not single-valued at $z=z_0$. (?)
Am I right? So, my question is actually whether both non-differentiability and not being a single-valued function both apply to this case. I know that either is enough to show non-analyticity.