Suppose $(\mathbb R^n,0)\overset{f}{\to}(\mathbb R^k,0) $ is a real analytic map with $0\in \mathbb R^n$ an isolated critical point. I have read in many places the following assertion. For sufficiently small $\varepsilon >0$ there exists $\delta>0$ such that fibers over $B_\delta(0)\subset \mathbb R^k$ have a transverse intersection with $\mathbb S_\varepsilon ^{n-1}(0)\subset \mathbb R^n$
First of all this seems like a consequence of the local (on the source) structure of (not necessarily analytic) $C^\infty$ submersions: upon deleting the critical point, the fibers are "aligned", so a sufficiently small sphere upstairs should intersect them transversally.
The problem is that "straightening fibers" can deform the sphere. Intuitively, a sufficiently small sphere won't be deformed too much, but I'm not sure how to write a proof of the assertion. Does it indeed hold in the $C^\infty$ case?