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For an open set $U$ we have that a complex function $f: U \to \mathbb{C} $ is analytic iff it satisfies the Cauchy-Riemann equations and it's partials are continuous. So for open sets the situation is clear to me.

I did some searching on this site and found that the situation for non-open subsets, say a point for example seems to be the same:

See here for example

If Cauchy-Riemann hold and the first order partials are continuous does that imply it is differentiable?

But then I also came across this questions:

Showing $f(z)=x^2+iy^3$ is not analytic anywhere

Here it seems to be that, although $f$ has continuous partials at the origin (they are polynomials in $x$ or $y$) and the Cauchy-Riemann equations are satisfied at the origin, $f$ is not holomorphic at $(0,0)$. Infact it is stated that the Cauchy-Riemann equations must be satisfied on an open subset containing the point.

Can someone clarify what's going on here?

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    Complex differentiability at a point only means that the Cauchy-Riemann equations hold at that point. Holomorphicity (analyticity) means that the Cauchy-Riemann equations hold in an open set. Holomorphicity is a much stronger condition than complex differentiability (it is complex differentiability on an open set). – Cameron Williams Dec 20 '19 at 16:17
  • Oh I see, so holomorphic $\neq$ complex differentiable!? –  Dec 20 '19 at 16:18
  • Correct. Complex differentiability says nothing about the set it occurs on. – Cameron Williams Dec 20 '19 at 16:18
  • I see, so the question "Is $f(x,y)= x^2+iy^3$ complex differentiable at$(0,0)$?" and "Is $f(x,y)=x^2+iy^2$ holomorphic at $(0,0)$?" have different answers... In a sense it then doesn't even make sense to talk about being holomorphic/analytic at a point? Is this correct? Many thanks! –  Dec 20 '19 at 16:21
  • Some authors will state that holomorphic at a point means that the function is actually holomorphic on an open set containing that point, but ultimately it's the same definition since it relies it being holomorphic on an open set. – Cameron Williams Dec 20 '19 at 16:23
  • Thanks! Just to make sure I've understood this properly: $x^2+iy^3$ is analytic nowhere because the C.R. equations fail on any openset- and for holomorphic we care about open sets. However $x^2+iy^3$ is complex differentiable at $(0,0)$ as the C.R. equations are satisfied at $(0,0)$ and have continuous partials - all we care about for complex differentiability is the single point ( in this case). Sorry to repeat, but have I understood this correctly? –  Dec 20 '19 at 16:28
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    Yep you nailed it! Good work :) – Cameron Williams Dec 20 '19 at 16:29

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