I am faced with the following problem:
Let $z_{0}\neq 0$ and let $f(z) = \ln r + i \varphi$, where $r = |z|$, $\varphi \in arg z$, and $\varphi$ is chosen so that $f$ is continuous in a neighborhood of $z_{0}$. Prove that $f$ is differentiable in a neighborhood of $z_{0}$.
Being differentiable in a neighborhood of $z_{0}$ means that it's analytic there, right? It sounds to me like the problem is asking me to show that the Cauchy-Riemann equations hold at $z_{0}$, but I'm not sure how the continuity of $f$ fits in here.
Could someone please let me know what exactly I am being asked to do here? I'm very confused and am not sure how to begin, other than just applying the Cauchy-Riemann equations in polar form, at least.