Suppose that $f$ is a holomorphic function function in a neighborhood of $z_0$, $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ exist at $z_0$. Must Cauchy–Riemann equation $\dfrac{\partial f}{\partial y} = \mathrm{i}\dfrac{\partial f}{\partial x}$ hold at the point $z_0$?
If we do not assume that $f$ is differentiable as a bivariate function at $z_0$, then $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ are not necessarily continuous with respect to $x$ or $y$, as shown by the example $f(z) = \begin{cases} z^2\sin\dfrac{1}{z^4}, &z\neq 0\\ 0, &z=0\end{cases}$. Of course, the equation can be violated only when $z_0$ is an essential sigularity of $f$. I can't currently come up with a proof or a counter-example. Any help appreciated.