Morse-Sard for Euclidean maps

Prove the assertion by induction on the source dimension $m$, keeping the differentiability order $r$ fixed. If $m=0$, then $U$ is at most a point, so $\mathrm{Crit}(f)$ and its image are finite, hence null in ${\mathbb{R}}^n$. Assume $m\ge 1$ and the theorem proved for all smaller source dimensions and every positive target dimension satisfying the same differentiability inequality. Exhaust $U$ by the canonically enumerated rational closed cubes with closures inside $U$. We use the following choice-free countable-union observation throughout: every compact null subset of ${\mathbb{R}}^n$ has, for each positive rational budget, a finite rational-cube cover below that budget. Indeed, start with a countable cube cover of total volume below half the budget from [F2], enlarge its cubes slightly to rational open cubes while retaining the budget, and take a finite subcover by compactness. Enumerate all finite rational-cube families and use the first qualifying family for each compact set and budget. For a countable sequence of compact null sets, use budgets $\epsilon 2^{-i-1}$ and combine these canonical finite covers; [F2] makes their union null without making countably many arbitrary choices. Thus it suffices to prove $f(\mathrm{Crit}(f)\cap Q_{\nu })$ null for each cube: these images are compact and their countable union is the critical value set of [F1].

Fix one cube $Q_{\nu }$ and put $C_0:=\mathrm{Crit}(f)\cap Q_{\nu },C_j:=\{x\in Q_{\nu }:D^{\alpha }f(x)=0\text{ for every }1\le \mathrm{\mid }\alpha \mathrm{\mid }\le j\}(j\ge 1)\mathrm{.}$ These are the intersections of $Q_{\nu }$ with the corresponding closed strata in $U$. Then $C_0=(C_0\setminus C_1)\cup {\bigcup }_{j=1}^{r-1}(C_j\setminus C_{j+1})\cup C_r\mathrm{.}$ For each $1\le j<r$ and each $\mathrm{\ell }\ge 1$, the set $K_{j,\mathrm{\ell }}:=\{x\in C_j:\mathrm{dist}(x,C_{j+1})\ge 1\mathrm{/}\mathrm{\ell }\}$ is compact and contained in $C_j\setminus C_{j+1}$ (take distance to the empty set as infinite). Because $C_j\setminus C_{j+1}={\bigcup }_{\mathrm{\ell }\ge 1}{K}_{j,\mathrm{\ell }},$ the nonflat lemma in [L1] applies: its lower-dimensional Sard premise holds by the induction hypothesis at the unchanged order $r$, since $r>\max \{m-n,0\}$ implies $r>\max \{(m-1)-n,0\}$. Each $f(K_{j,\mathrm{\ell }})$ is compact null, so the canonical-cover observation in step 1.1 makes $f(C_j\setminus C_{j+1})$ null. Also, $C_r$ is compact, and $r>\max \{m-n,0\}$ implies $rn\ge m$; the flat lemma in [L1] makes $f(C_r)$ null.

It remains to show that $f(C_0\setminus C_1)$ is null. If $n=1$, then a linear map ${\mathbb{R}}^m\to \mathbb{R}$ is surjective exactly when it is nonzero, so $C_0=C_1$ and there is nothing to prove. Assume $n>1$, and fix $x\in C_0\setminus C_1$. Some first partial derivative of some component of $f$ is nonzero at $x$; after reordering coordinates and components, assume $\frac{\mathrm{\partial }{f}^1}{\mathrm{\partial }{x}^1}(x)\mathrm{\ne }0.$ By [L2], after shrinking choose a neighbourhood $W_x$ of $x$ and a $C^r$ diffeomorphism ${\mathrm{\Phi }}_x(y):=(f^1(y),y^2,\dots ,y^m)$ from $W_x$ onto an open set $I_x\times {\mathrm{\Omega }}_x\subseteq \mathbb{R}\times {\mathbb{R}}^{m-1}$. Write $f\circ {\mathrm{\Phi }}_x^{-1}(t,u)=(t,{\tilde{f}}_x(t,u))\mathrm{.}$ Each slice map $u\mapsto {\tilde{f}}_x(t,u)$ is $C^r$, and because $r>\max \{m-n,0\}=\max \{(m-1)-(n-1),0\},$ the induction hypothesis applies to those maps. If $q={\mathrm{\Phi }}_x^{-1}(t,u)\in C_0\cap W_x$, then in these coordinates the differential of $f$ has block form $D{f}_q=\left[\begin{array}{cc}1& 0\\ \ast & D({\tilde{f}}_x{)}_t(u)\end{array}\right],$ so $q$ is critical for $f$ exactly when $u$ is a critical point of the slice $u\mapsto {\tilde{f}}_x(t,u)$. For every rational open cube $B$ whose closure lies in $W_x$, the compact set $f(C_0\cap \bar{B})$ has sections contained in the slice critical value sets, hence null in ${\mathbb{R}}^{n-1}$ by induction. The slicing lemma in [L1] makes this compact image null in ${\mathbb{R}}^n$. Consider all rational cubes $B$ for which such an $x$ and chart exist; they form a countable family covering $C_0\setminus C_1$. Each has the compact-null conclusion just proved, independently of which witnessing chart exists. The canonical-cover observation in step 1.1 makes their image union $f(C_0\setminus C_1)$ null without selecting charts or covers countably.

Step 2.1 shows that $f(C_j\setminus C_{j+1})$ is null for every [F1, step 2.1, step 2.2, step 1.1] $1\le j<r$ and that $f(C_r)$ is null, while step 2.2 handles $f(C_0\setminus C_1)$. Hence $f(C_0)=f(\mathrm{Crit}(f)\cap Q_{\nu })$ is null. Applying step 1.1 shows that the whole critical value set of $f$ is null.