Assume $f:\mathbb{C} \to \mathbb{C}$ is $\mathbb{R} $-linear function. Assume in addition that there exists $z_0 $ where $ f $ preserves angle, that is, for any regular curves $ \gamma\left(t\right),\delta\left(t\right) $, that are intersecting in a point $ z_0 $, we have $$ \angle\left(\gamma,\delta\right)|_{z=z_{0}}=\angle\left(f\left(\gamma\right),f\left(\delta\right)\right)|_{z=f\left(z_{0}\right)} $$
How can I prove that $ f $ is also $\mathbb{C}$- linear?
The original question was to prove that both directions are correct. I proved that if $ f $ is $\mathbb{C} $ linear then it is also preserves angles.Im not sure how to prove the other direction.
Any help would be appreciated. Thanks