= Solution
Complex differentiability requires the same limit for increments along the real and imaginary directions. Real increments give $f'=u_x+iv_x$. Imaginary increments give
$$
f'=\frac{u_y+iv_y}{i}=v_y-iu_y.
$$
Equating real and imaginary parts yields the <Cauchy-Riemann equations>
$$
\boxed{u_x=v_y,\qquad u_y=-v_x.}
$$
They hold throughout the open region where $f$ is <holomorphic>. Conversely, continuous first real <partial derivatives> satisfying these equations on an open set ensure holomorphy there.
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