Suppose that $f:\mathbb{R}^n\to \mathbb{R}^m$ is of class $C^1$ and $Df(x_0)$ has rank $m$. Then show there is a whole neighborhood of $f(x_0)$ lying in the image of $f$.

My attempt: if $Df(x_0)$ is onto (rank $m$) and $n\leq m$, then I can use the Rank Theorem and justify that exists open sets $V,W\subset \mathbb{R}^m$, $f(x_0)\in V$ and $\psi:V\to W$ such that: $$(\psi \circ f)(x_1,\cdots,x_n)=(x_1,\cdots,x_n,0,\cdots,0)$$ Particularly, its mean that $V\subset Im(f)$.

Is this correct? What happend when $n>m$?

Thanks for your help.


if $n<m$ then the rank cannot be $m$ (linear algebra), if $n\geqslant m$ then your local representation $$ (x_1,\dots,x_n) \mapsto (x_1,\dots,x_m) $$ proves the assertion


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