I'm reading the proof of a lemma in Milnor's Topology from the Differentiable Viewpoint, specifically Lemma 4 of Chapter 2. I am caught up on a detail. Essentially, it amounts to the following:
Let $f:R^m\to R^n$ be smooth and let $y\in R^m$ be a regular value. Let $x\in f^{-1}(y)$. Is it true that there is some neighborhood $U$ of $x$ such that $f$ has no critical points in $U$?
Yes, this is true.
Recall that the rank is lower semi-continuous, so that locally $\mathrm{rk}(\mathrm{d}_xf)$ can only grow, but since $x$ is a regular point, this rank is maximal and there is an open neighbourhood of $x$ constituted only of regular points of $f$.
Reminder. Let $A$ be a matrix of rank $r$, therefore there exists $I,J\subset\{1,\ldots,n\}$ of cardinality $r$ such that: $$A_{I,J}:=(a_{i,j})_{i\in I,j\in J}$$ is invertible i.e. $\det(A_{I,J})\neq 0$, but $\det$ is a polynomial map with respect to the entries and is continuous. Hence, for $B$ in a neighborhood of $A$, $\det(B_{I,J})$ does not vanish and $B$ has at least rank $r$.