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 $u$?