= Solution
Under the assumed continuous real differentiability, the necessary and sufficient conditions are the <Cauchy-Riemann equations> at every point of $U$:
$$
\boxed{u_x=v_y,\qquad u_y=-v_x.}
$$
Necessity follows by computing the complex <derivative> along real and imaginary increments. For sufficiency, real differentiability gives the first-order expansion
$$
f(z+h)-f(z)=(u_x+iv_x)\operatorname{Re}h+(u_y+iv_y)\operatorname{Im}h+o(|h|).
$$
The <Cauchy-Riemann equations> turn the linear part into $(u_x+iv_x)h$, so the complex <derivative> exists and equals $u_x+iv_x$. Thus $f$ is a <holomorphic function> on $U$.
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