Joint continuity of division of smooth functions. II. The distance to a Whitney stratified set from a transversal submanifold

Abstract:

Let $S$ be a closed set in ${{\mathbf {R}}^m}$, and let a ${C^1}$ Whitney stratification of $S$ be given. (Actually, only Whitney (a)-regularity is needed.) Let $f:{{\mathbf {R}}^n} \to {{\mathbf {R}}^m}$ be a ${C^1}$ map transversal to all the strata. Assume that the image of $f$ intersects $S$. Then for each compact set $K$ in ${{\mathbf {R}}^n}$, the Euclidean distances $\rho (x,{f^{ - 1}}(S))$ and $\rho (f(x),S)$, for $x$ in $K$, are bounded by constant multiples of each other. The bounding constants can be chosen to work for all maps $g$ which are close enough to $f$ in a ${C^1}$ sense on a neighborhood of $K$. This result is used in part I (written jointly with S. Shnider) to prove a result on the joint continuity of the division of smooth functions [MS].