= Solution
For a regular smooth curve $z=z_j(t)$ passing through $z_0$, the complex tangent of its image is $dw_j/dt=f'(z_0)z_j'(t)$. When $f'(z_0)\ne0$, both tangents are multiplied by the same nonzero complex number. Multiplication rescales lengths by $|f'(z_0)|$ and rotates every direction by $\arg f'(z_0)$, so the difference of tangent arguments is unchanged. Hence the <analytic function> defines a <conformal map> at $z_0$: \b[oriented angles, and therefore ordinary angles between the curves, are preserved]. The nonzero derivative condition also ensures that the image tangents are nonzero.
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