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I'm reading a pdf where it's said that the function ##f: \mathbb R \longrightarrow \mathbb{R}^2## given by ##f(x) = \langle \sin (2 \pi x), \cos ( 2 \pi x) \rangle## is not one-to-one, because ##f(x+1) = f(x)##. This is pretty obvious to me. What I don't understand is that next they say that the derivative map, which they denote by ##Df(x)##,

Is'nt the derivative given by ##2 \pi \langle \cos (2 \pi x), - \sin (2 \pi x) \rangle##? It's easy to see that it's the same for ##x## and ##x + 1##.

**is injective for all**##x \in \mathbb R##. How can it be?Is'nt the derivative given by ##2 \pi \langle \cos (2 \pi x), - \sin (2 \pi x) \rangle##? It's easy to see that it's the same for ##x## and ##x + 1##.

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