= Solution
For an <analytic function>, the <complex differentiability> limit is independent of the direction of approach. A real increment gives $f'(z)=u_x+iv_x$, whereas an imaginary increment gives
$$
f'(z)=\frac{u_y+iv_y}{i}=v_y-iu_y.
$$
Equality of real and imaginary parts proves the <Cauchy-Riemann equations>:
$$
\boxed{u_x=v_y,\qquad u_y=-v_x.}
$$
Conversely, if $u,v$ have continuous first partial <derivatives> near a point and obey these equations, their real differentiability gives, for a complex increment $h$,
$$
f(z+h)-f(z)=(u_x+iv_x)\operatorname{Re}h+(u_y+iv_y)\operatorname{Im}h+o(|h|)=(u_x+iv_x)h+o(|h|).
$$
Dividing by $h$ proves the <complex differentiability> there. Continuous first partial <derivatives> are a sufficient regularity hypothesis for this converse; agreement along the two coordinate directions alone is only a necessary condition.
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