Suppose that $m < n$, that $U$ is an open set in $\mathbb{R}^m$ and that $f:U \to \mathbb{R}^n $ is a $C^1$ function that has maximal rank (rank $m$) everywhere in $U$. Show that, for each $x \in U$, there is a neighbourhood $V$ of $f(x)$ and a $C^1$ function $g: V \to \mathbb{R}^m$ that is a left inverse of $f$ in $U$.
Is this simply a direct consequence of the Rank Theorem, i.e. we can find an open set and coordinates such that $f$ is the map $i(x^1, ..., x^m)=(x^1, ..., x^m, 0, ..., 0) \in \mathbb{R}^n$. Or am I misunderstanding the theorem?
Would it be better to apply the inverse function theorem and implicit function theorem?