Let $a$ be in a point in $\mathbb{R}^m$.
$f$ is a partial function from $\mathbb{R}^m$ to $\mathbb{R}^n$, where $m\geq n$, that is differentiable in a neighbourhood $V$ of $a$ and continuously differentiable at $a$.
I'm trying to prove that if $J_{f_a}$ has full rank then $f(V)$ is a neighbourhood of $f(a)$.
I figured that if $m=n$ then this means that $D_{f_a}$ is invertible and the proof is obtained via the inverse function theorem. Assuming this is correct, then to handle the case where $m>n$ I would have to form some other function from $f$ so that I can again use the inverse function theorem somehow. The problem is I simply cannot see what kind of function I can form.
I even tried considering a concrete example with $(x,y,z) \mapsto (x+y,z)$ to see which kind of function would be helpful here (maybe $(x+y,z,x)$?), but I'm struggling to generalise this.
I also thought of somehow using the implicit function theorem because of the similar hypotheses and that it allows for dimensionality reduction, but I'm not really seeing how it could be of help.
What am I missing?
