= Solution
A <conformal map> at a point $z_0$ is a map that preserves oriented angles between regular differentiable curves through $z_0$. In the differential formulation, it is real differentiable at $z_0$ and its derivative is an orientation-preserving similarity: a rotation followed by multiplication of lengths by a positive constant. The map is conformal on $U$ when it has this property at every point:
$$
\boxed{D f(z_0)=\lambda R_\alpha,\qquad\lambda>0}.
$$
Here $R_\alpha$ is a planar rotation. Conformality is a local property and does not itself require global injectivity. We use the oriented convention standard in <complex analysis>; a reflection reverses oriented angles.
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