existence of analytic continuation

Question

Good morning,

I have just started reading Riemann surfaces. I would like to ask a question, maybe it is naive.

Let $X$ be a Riemann surface and $\phi\in\mathcal{O}_{a,X}$ a holomorphic function germ at $a$ of $X.$ Let $u : [0,1]\to X$ be a curve, i.e a continuous mapping. Does it exist always an analytic continuation of $\phi$ along the curve $u$?

Answer 1

No, e.g. you may run into a singularity. For example, take $X = {\mathbb C}$, $u(t) = t$, $a=0$ and $\phi(z) = \frac{1}{1-2z}$ in a neighbourhood of 0. The pole at $t = 1/2$ stops the analytic continuation along the curve.