I just noticed in the rule about the sufficient conditions for complex differentiability via Cauchy-Riemann:
When we're considering as to whether or not $g: \mathbb C \to \mathbb C$ is differentiable at $z=z_0$, we don't need $g_x,g_y$ to be continuous on the entire neighbourhood of the point $z_0$. Apparently, all we need is that $g_x,g_y$ exist on the neighbourhood and are continuous merely at $z=z_0$ (and satisfy Cauchy-Riemann at $z=z_0$).
Is this correct?
Please give any example (preferably as simple as possible) where for $g: G \to \mathbb C$ ($G$ is a subset of $\mathbb C$), for $z_0 \in G$ and for open neighbourhood $U$ of $z_0$ with $U \subseteq G$, we have that $g_x$ and $g_y$ exist but are not continuous on the entire $U \ \setminus \ \{z_0\}$ but are still continuous at $z_0$ and satisfy Cauchy-Riemann at $z=z_0$. There are probably some simple (yet difficult) examples that are obvious that I've missed, but please tell me anyway.