Is the function $$ W(y) = \int_{f^{-1}(\{y\})} \phi(x) \mathcal{H}^{n-m}(dx) $$ continuous over regular values $y\in\mathbb{R}^m$, for a $C^\infty$-smooth function $f:\mathbb{R}^n\to\mathbb{R}^m$ with $n > m$ and suitably smooth $ \phi$?
- I am aware of this question, but there $f$ was only continuous and $y$ didn't need to be a regular value.
- Notice that by Sard's theorem $\mathcal{H}^{n-m}(fA_m)= 0 $ where $A_m$ is the set of critical points in $\mathbb{R}^n$ such that the rank of the differential there is less than $m$.