Holomorphic function at a point

Question

A map $f:X\to Y$ between Riemann surfaces is called holomorphic if for every complex charts $\varphi $ and $ \psi$ (of $X$ and $Y$ resp) we have that $\psi \circ f \circ \varphi^{-1}$ is holomorphic. I would like to know if there is a local definition of this i.e. I would like to know if given a map $f:X\to Y$ between Riemann surfaces and $ x\in X$ a point is there a notion of $f$ being holomorphic at $x$. Thank you all in advance

Answer 1

Yes, it's just the obvious thing: $f$ is holomorphic at $x$ if whenever $\varphi$ is a complex chart of $X$ defined at $x$ and $\psi$ is a complex chart of $Y$ defined at $f(x)$, $\psi\circ f\circ\varphi^{-1}$ is holomorphic at $\varphi(x)$.