Is the function injective if the Jacobian has full column rank?

Let $f:mathbb{R}^m to mathbb{R}^n: x to f(x)$ be a continuous and differentiable function with $m < n$. If the Jacobian $J_f$ has full column rank (i.e., rank=$m$) $forall x in mathbb{R}^m$, does this imply that $f$ is an injective function? If yes, can I get a reference for this result?

Mathematics Asked by Abdul Fatir on December 30, 2020

1 Answers

One Answer

No, take $f(t) =pmatrix{ sin t\ cos t}$.

Correct answer by daw on December 30, 2020

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