= Solution
At a point $z_0=x+iy$, <complex differentiability> means that
$$
f'(z_0)=\lim_{h\to0,\ h\in\mathbb C\setminus\{0\}}\frac{f(z_0+h)-f(z_0)}h
$$
exists and is independent of the direction of approach. Write $f=u+iv$. Restricting to real $h$ gives $f'=u_x+iv_x$. Restricting to $h=it$, with real $t\to0$, gives
$$
f'=\frac{u_y+iv_y}{i}=v_y-iu_y.
$$
These directional limits establish existence of the displayed partial derivatives. Equating their real and imaginary parts gives \b[the Cauchy-Riemann equations]
$$
\boxed{u_x=v_y,\qquad u_y=-v_x.}
$$
The argument applies at every point of the given open domain; it does not assume continuity of the partial derivatives beforehand.
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