# Holomorphic function at a point

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

• The notion of “holomorphic” isn’t applied to points, but instead “complex differentiable”. Then you can define a function of Riemann surfaces $f \colon X → Y$ to be differentiable at $x$, if for any charts $φ$ on $X$ and $ψ$ on $Y$, $ψ ∘ f ∘ φ^{-1}$ is complex differentiable at $φ(x)$. May 29, 2019 at 22:33
• @k.stm: Typically "holomorphic at a point" is defined to mean "complex differentiable in a neighborhood of the point". May 29, 2019 at 22:35
• @EricWofsey Ah, yeah. Makes sense. I like that. May 29, 2019 at 22:36
• @EricWofsey If you say so. I'd restrict "holomorphic" to open sets; the standard meaning of "analytic on E" is "holomorphic in some nbd of E". May 29, 2019 at 23:11

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 $$\varphi(x)$$.
• Nope. I'm not sure I've ever actually seen it used, since typically the only reason to be interested in a function holomorphic at a point is for studying local properties and so there's no reason to talk about Riemann surfaces rather than just $\mathbb{C}$. It's the obvious definition though and can be considered as a special case of the very general principle that any local property on Euclidean space gives rise to a corresponding local definition on manifolds. May 29, 2019 at 23:35
• I have never seen "analytic" and "holomorphic" distinguished the way you say. I have absolutely seen holomorphic functions on non-open subsets of $\mathbb{C}$ though (I just don't specifically recall seeing it more generally on Riemann surfaces). For instance, Stein and Shakarchi's Complex Analysis defines holomorphic functions at a single point of $\mathbb{C}$, or more generally on closed subsets of $\mathbb{C}$. May 30, 2019 at 0:03
• " the very general principle that any local property on Euclidean space gives rise to a corresponding local definition on manifolds" hmm, perhaps I haven't been clear. Yes of course if one did speak of "holomorphic at $x$" in a Euclidean context then the extension to Riemann surfaces would be exactly as you say. I'm asserting that there iis no "holomorphic at $x$" in $\Bbb C$. May 30, 2019 at 0:04