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I've been thinking about a problem in Spivak's Calculus on Manifolds and noticed that it can be proven quite cleanly if the following is true:

Let g:R^n->R^n be a differentiable 1-1 function. Then we can find a point s.t. det g'(x) != 0.

Geometrically this means that the best linear approximation for the function at at least one point is injective. However I can't seem to be able to lift this up to mean non-injectivity of the function itself if the linear approximation is always non-injective. I don't even know whether or not this is true.

Anyone know whether this is true or not and how to prove it or give a counter example?

Thanks.