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

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

Correct answer by daw on December 30, 2020

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