Sard on the infinitely flat critical stratum

If $m=0$, then $K$ has at most one point and $f(K)$ is finite, hence null in ${\mathbb{R}}^n$ by cubes of arbitrarily small side. Assume henceforth that $m\ge 1$. For each $x\in K$, choose nested closed cubes $Q_x\subseteq {\widehat{Q}}_x\subseteq U$ with $x\in \mathrm{int}(Q_x)$ and ${\widehat{Q}}_x$ having $Q_x$ in its interior. Compactness gives finitely many inner cubes $Q_1,\dots ,Q_s$ whose interiors cover $K$. It suffices to prove that each $f(K\cap Q_i)$ is null, because finite unions of Euclidean null sets are null directly from the cube-cover definition [F1].

Fix one pair $Q\subseteq \widehat{Q}$, let $K_Q:=K\cap Q$, and let $\lambda$ be the side length of $Q$. The finitely many order-$r$ partial derivatives of the components of $f$ are uniformly continuous on the compact cube $\widehat{Q}$ by [L2]. Since they vanish at every $x\in K_Q$, applying [L1] componentwise with degree $r-1$ shows that for every $\eta >0$ there is a single $\delta >0$ such that $\mathrm{\mid }f(y)-f(x)\mathrm{\mid }\le \eta \mathrm{\parallel }y-x{\mathrm{\parallel }}^r$ whenever $x\in K_Q$, $y\in Q$, and $\mathrm{\parallel }y-x\mathrm{\parallel }<\delta$.

Fix $\epsilon >0$. If $rn=m$, choose $\eta >0$ so small that $(2\eta {)}^n{m}^{rn\mathrm{/}2}{\lambda }^{rn}<\epsilon$; if $rn>m$, choose any $\eta >0$. Let $\delta$ be furnished by step 2.1, and choose $N$ so large that the congruent subcubes in the subdivision of $Q$ into $N^m$ cubes have diameter below $\delta$. Use the following target cubes. [step 2.1, choose, cases] For each subcube $Q_{\nu }$ meeting $K_Q$, choose a point $x_{\nu }\in K_Q\cap Q_{\nu }$. If $y\in K_Q\cap Q_{\nu }$, then $\mathrm{\parallel }y-x_{\nu }\mathrm{\parallel }\le \sqrt{m}\lambda \mathrm{/}N$, so step 2.1 gives $\mathrm{\mid }f(y)-f(x_{\nu })\mathrm{\mid }\le \eta (\frac{\sqrt{m}\lambda }{N}{)}^r\mathrm{.}$ Hence $f(K_Q\cap Q_{\nu })$ lies in an $n$-cube of side length at most $2\eta (\frac{\sqrt{m}\lambda }{N}{)}^r\mathrm{.}$

The union of those target cubes covers $f(K_Q)$, and its total $n$-volume has the following bound. [F1, step 3.1, cases, algebra] It is at most $N^m\left(2\eta \right(\frac{\sqrt{m}\lambda }{N}{)}^r{)}^n=(2\eta {)}^n{m}^{rn\mathrm{/}2}{\lambda }^{rn}{N}^{m-rn}\mathrm{.}$ If $rn>m$, increase $N$ until this quantity is below $\epsilon$; if $rn=m$, the choice of $\eta$ in step 3.1 already makes it smaller than $\epsilon$. In either case the total covering volume is below $\epsilon$. Therefore [F1] implies that $f(K_Q)$ is null.

Applying step 4.1 to the finite cover from step 1.1 shows that $f(K)$ is null. Thus the infinitely flat critical stratum has null image.