What do the Cauchy-Riemann equations tell us about being differentiable on a non-open subset?

Question

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?