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Does there exist an infinite family of functions which satisfy

$|f^\prime(x)|=1$ and $f(1)=f(-1)=0$?

where a) $f\colon \mathbb{R}\to \mathbb{R}$ b) $f$ is a complex function defined on some open neighbourhood of the closed unit disc in the plane.

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$|f'(x)| = 1$ everywhere? At one point? Where exactly? – Ayman Hourieh Jan 14 at 21:31
Yes, constantly as a complex function. – Drake Jan 14 at 21:34
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Can you answer a)? – Jonas Meyer Jan 14 at 21:35
So you're interested if its true in case a) or in case b)? – JSchlather Jan 14 at 21:39
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You should have an idea at least of what goes wrong with (a) by trying to sketch a picture. For (b), because of existence of "$f'(x)$" I guess you are talking about complex differentiability. For such $f$, $f'$ is also complex differentiable. Do you know which complex differentiable functions (on a connected open set) have constant modulus? – Jonas Meyer Jan 14 at 21:44
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