The definition of an "isogonal mapping" in $R^2$ is as follows.
Let $D$ be a domain in $R^2$. An isogonal mapping $f:D\to R^2$ is a transformation that preserves the magnitudes of local angles, but not their orientation.
Question: A continuous isogonal mapping $f:D\to R^2$ is necessarily holomorphic or anti-holomorphic on $D$?
I believe the answer should be known "yes" but I donot find the proof in a suitable literature. I wish to know how to prove it.
EDIT. $f$ maps $C^1$ curves to $C^1$ curves, then "$f$ preserves the magnitudes of local angles" is well-defined as well as the angles. We donnot assume that $f$ is differentiable in advance.