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

Question

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?

Answer 1

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