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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.

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You should clarify what it means to "preserve the magnitudes of local angles" when $f$ is not necessarily differentiable. For differentiable maps this question has been answered. –  user53153 Jan 23 '13 at 19:55
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