Is it true that a c¹ map with everywhere non-zero jacobian is injective?

Question

Accepted Answer

No. f(x,y) = (eˣ cos y, eˣ sin y) on ℝ² The Jacobian determinant is e^{2x} ≠ 0, but f(x, y) = f(x, y + 2π). Invertibility is only local.

$A C^{1}$ map with everywhere non-zero Jacobian is injective