A clipped affine function keeps its zero region

The declared assumption is the Axiom of Choice [F1]; by [F2] Countable Choice holds, so the choice hypotheses of [F4], [F13], [F17] and [F19] are in force, while [F5] is stated under the declared Axiom of Choice itself. Since $M>0$, the cube is nonempty: $-2M<0<2M$, so $0\in Q$. No representative is selected in this proof.

Elementary clipping identities. For every real $t$ one has $\max \{0,t\}=0$ for $t\le 0$ and $\max \{0,t\}=t$ for $t\ge 0$ by [F21] and [F22]; consequently, for $M>0$, $\min \{M,\max \{0,t\}\}=0$ for $t\le 0$, $=t$ for $0<t<M$, and $=M$ for $t\ge M$. Moreover $\max \{-M,\max \{0,t\}\}=\max \{0,t\}$ for every $t$: with $u:=\max \{0,t\}$ one has $u\ge 0$ and $-M<0\le u$ by [F22], so $u$ is the larger of the two arguments [F21]. Hence the truncation of [F5] satisfies $T_M(\max \{0,t\})=\min \{M,\max \{-M,\max \{0,t\}\}\}=\min \{M,\max \{0,t\}\}$ for every $t$.

The coordinate function $f(x):=x_1$ is of class $C^1$ on $Q$, with ${\mathrm{\partial }}_{1}f\equiv 1$ and ${\mathrm{\partial }}_{j}f\equiv 0$ for $j\ge 2$. Fix $a\in Q$ and a coordinate index $j$. For real $t\mathrm{\ne }0$, [F8] and the pointwise operations give $f(a+t{e}_j)=(a+t{e}_j{)}_1=a_1+t(e_j{)}_1$, so the difference quotient of [F7] is $\frac{f(a+t{e}_j)-f(a)}{t}=\frac{a_1+t(e_j{)}_1-a_1}{t}=(e_j{)}_1$ for every $t\mathrm{\ne }0$ by [F22]; this is the constant function $t\mapsto (e_j{)}_1$ on $\mathbb{R}\setminus \{0\}$. For every $\epsilon >0$ the choice $\delta :=1$ gives $\mathrm{\mid }(e_j{)}_1-(e_j{)}_1\mathrm{\mid }=0<\epsilon$ whenever $0<\mathrm{\mid }t\mathrm{\mid }<\delta$, so the limit of the difference quotient at $0$ is $(e_j{)}_1$ in the sense of [F12], and ${\mathrm{\partial }}_{j}f(a)=(e_j{)}_1$ exists by [F7]; by [F8] this value is $1$ for $j=1$ and $0$ for $j\ge 2$. The function $f$ is the first coordinate projection, hence continuous on $Q$ by [F9], which is the $\epsilon$-$\delta$ notion by [F10] and [F11]; and each ${\mathrm{\partial }}_{j}f$ is a constant function on $Q$, hence continuous by the $\epsilon$-$\delta$ condition of [F11], any $\delta >0$ serving at every point. Therefore every word of length at most $1$ in the sense of [F6] has an existing continuous iterated derivative, that is, $f\in C^1(Q)$.

Bounds and $L^p$ membership of $f$ and its first partials. The cube $Q$ is the box $R^{\circ }$ of [F13] with $a_i=-2M$ and $b_i=2M$, so it is open and ${\lambda }_n(Q)=(4M{)}^n<\mathrm{\infty }$, and for $x\in Q$ the inequalities $-2M<x_1<2M$ give $\mathrm{\mid }{x}_1\mathrm{\mid }<2M$ by [F20]. Hence $\mathrm{\mid }f\mathrm{\mid }\le 2M$ on $Q$, while $\mathrm{\mid }{\mathrm{\partial }}_{1}f\mathrm{\mid }=1$ and ${\mathrm{\partial }}_{j}f=0$ on $Q$ by step 1.3. The functions $f$, ${\mathrm{\partial }}_{1}f$ and ${\mathrm{\partial }}_{j}f$ are continuous on $Q$ by step 1.3, hence Borel measurable and Lebesgue measurable under Countable Choice by [F17]. Therefore $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\le 2M$ and $\mathrm{\parallel }{\mathrm{\partial }}_{j}f{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ by [F15], so the classes of $f$ and of each ${\mathrm{\partial }}_{j}f$ lie in $L^{\mathrm{\infty }}(Q)$ by [F16], and in ${\mathcal{L}}^p(Q)$ for every $1\le p<\mathrm{\infty }$ by [F14] applied to the finite measure ${\lambda }_n(Q)$; for $p=\mathrm{\infty }$ the membership is the $L^{\mathrm{\infty }}$ statement itself.

Region values (clause 2 of the Statement). By step 1.2 applied at $t=x_1$, the function $w(x)=\min \{M,\max \{0,x_1\}\}$ vanishes for $x_1\le 0$, equals $x_1$ for $0<x_1<M$, and equals $M$ for $x_1\ge M$; these are pointwise identities on all of ${\mathbb{R}}^n$.

Classical derivatives are weak derivatives here: by [F4] applied with $k=1$ to the real-valued $C^1$ function $f$ of step 1.3, the classical derivatives ${\mathrm{\partial }}^{\alpha }f$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le 1$ are locally integrable and are the weak derivatives $D^{\alpha }f$; for $\alpha =0$ this says $D^{0}f=f$, and for $\alpha =e_j$ it says $D_{j}f={\mathrm{\partial }}_{j}f$ weakly. By step 2.1 the classes $[f]$ and $[{\mathrm{\partial }}_{j}f]$ lie in $L^p(Q)$ for every $1\le p\le \mathrm{\infty }$, so the definition [F3] gives $[f]\in W^{1,p}(Q;\mathbb{R})$ with $D_1[f]=[1]$ and $D_j[f]=[0]$ for $j\ge 2$, the classes of the constant functions $1$ and $0$ on $Q$.

The positive part. Let $u:=f^{+}$ be the positive part of the real class $[f]$ of step 3.1, as in [F23]; it is represented by the continuous function $\max \{0,x_1\}$ on $Q$. By [F5] applied to $[f]$ one has $u\in W^{1,p}(Q)$ with $D_{j}u=1_{\{f>0\}}{D}_{j}f$ almost everywhere on $Q$. Substituting the representatives of step 3.1 and the pointwise identity $\{f>0\}=\{x_1>0\}$, this says $D_{1}u=[{\mathbf{1}}_{\{x_1>0\}}]$ and $D_{j}u=[0]$ for $j\ge 2$ almost everywhere on $Q$.

The truncation. Apply [F5] again, now to the real class $u\in W^{1,p}(Q;\mathbb{R})$ of step 4.1 and the same level $M>0$: the truncation $T_{M}u$ lies in $W^{1,p}(Q)$ with $D_j{T}_{M}u=1_{\{\mathrm{\mid }u\mathrm{\mid }<M\}}{D}_{j}u$ almost everywhere on $Q$. By step 1.2 the representative $\max \{0,x_1\}$ of $u$ satisfies $T_M(\max \{0,x_1\})=\min \{M,\max \{0,x_1\}\}=w$ pointwise on $Q$ (indeed on ${\mathbb{R}}^n$), and $w$ is measurable: it is obtained from the measurable function $\max \{0,x_1\}$ by composition with the continuous, hence Borel, map $t\mapsto \min \{M,\max \{-M,t\}\}$ [F18]. So $w$ is a representative of the class $T_{M}u$, and hence $[w]=T_{M}u\in W^{1,p}(Q)$ for every $1\le p\le \mathrm{\infty }$. This proves clause 1 of the Statement.

Derivative computation. On $Q$ the level set of $u$ is $\{\mathrm{\mid }u\mathrm{\mid }<M\}=\{x_1<M\}$: since $u=\max \{0,x_1\}\ge 0$, the inequality $u<M$ holds when $x_1\le 0$ because then $u=0<M$, and when $x_1>0$ it reads $x_1<M$; conversely $x_1<M$ gives $u<M$. Consequently ${\mathbf{1}}_{\{\mathrm{\mid }u\mathrm{\mid }<M\}}={\mathbf{1}}_{\{x_1<M\}}$ pointwise on $Q$, and multiplying the representatives of step 4.1 gives ${\mathbf{1}}_{\{x_1<M\}}{\mathbf{1}}_{\{x_1>0\}}={\mathbf{1}}_{\{0<x_1<M\}}$ and ${\mathbf{1}}_{\{x_1<M\}}\cdot 0=0$ pointwise on $Q$ by [F21], [F22]; the two indicators are measurable because each threshold set is Borel and its indicator is Borel measurable (every inverse image is empty, the whole line, the threshold set, or its complement); composition with the measurable coordinate function and multiplication preserve measurability by [F18], so the displayed functions are legitimate representatives of the corresponding $L^p$ classes. Therefore $D_1[w]=[{\mathbf{1}}_{\{0<x_1<M\}}]$ and $D_j[w]=[0]$ for $j\ge 2$, all equalities almost everywhere on $Q$: the weak gradient of $w$ is represented by the vector field $g$ with $g(x)=e_1$ on the middle slab $\{0<x_1<M\}$ and $g(x)=0$ for $x_1\le 0$ and for $x_1\ge M$.

Interface irrelevance (clause 3 of the Statement). The pieces $Q\cap \{x_1=0\}$ and $Q\cap \{x_1=M\}$ are boxes with a degenerate side: in the terminology of [F13] each is contained in the closed box $\bar{R}$ with the first pair of endpoints both equal to $0$, respectively both equal to $M$, and the other pairs $(-2M,2M)$. For either box $R^{\circ }=\mathrm{\varnothing }$, and $R^{\circ }\subseteq Q\cap \{x_1=c\}\subseteq \bar{R}$ for the corresponding $c=0$ or $M$; hence [F13] gives measurability and measure zero. By [F19], modifying a locally integrable representative on a null set neither changes the weak-derivative relation nor the $L^p$ class of the derivative. Consequently any function that agrees with ${\mathbf{1}}_{\{0<x_1<M\}}$ off the two interface hyperplanes represents the same class $D_1[w]$, and any function that agrees with $0$ off them represents $D_j[w]$ for $j\ge 2$: the interface values are irrelevant, and the same conclusion is recorded by the level-set clause of [F5] at $\{u=M\}$.

Degenerate cases and accounting. The one-dimensional case $n=1$ is included: no step uses more than one coordinate direction, and $e_1$ is then the single standard basis vector. The exponent endpoints $p=1$ and $p=\mathrm{\infty }$ are included: steps 2.1 and 3.1 give $L^p$ membership of $f$ and its first partials at both endpoints, and [F5] is stated for every $1\le p\le \mathrm{\infty }$. The level is fixed at $M>0$, so $Q$ has side $4M>0$ and is nonempty by step 1.1; no limiting case $M\to 0$ or $M=\mathrm{\infty }$ is claimed, and clause 1 concerns the fixed cube $Q$. The only choice principle used is the Axiom of Choice, through [F2] for the classical-derivative lemma, the box-measure theorem, Borel-to-Lebesgue measurability and representative independence, and directly as the hypothesis of [F5]; no representative is selected, the argument at a general point selects only finitely many objects, and no Countable Choice beyond [F2] and no Dependent Choice is invoked. ∎