Monotone functions of two variables

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

By Proposition 3.3 of Rank coordinates, in the chart given by two windows every other rank coordinate is a function that is monotone in each of the two variables. Such functions are differentiable almost everywhere: Saks proved this in the plane, and Chabrillac and Crouzeix stated it in all dimensions [BBL, Theorem 1(f) and Section 3]. This lesson proves it, following OpenAI's argument [OpenAI-HL, Section 4]: monotonicity bounds oscillations by increments, a covering argument makes the increments linear in the scale at almost every point, and Rademacher's theorem applies on the pieces where this happens. The lesson also proves that a map of the plane has a null set of critical values, among the points where it is differentiable, without any further regularity. Both statements are used in Independent gradients and tangent directions.

From the core course Measure and Integration we use: Lipschitz functions on subsets of \(\mathbb R^2\) extend to Lipschitz functions on \(\mathbb R^2\) [Fremlin, 262B(b)]; Rademacher's theorem, that a Lipschitz function on \(\mathbb R^2\) is differentiable almost everywhere [Fremlin, 262Q]; and Lebesgue's density theorem, that almost every point of a measurable set \(E\) satisfies \(|E\cap B(x,\delta)|/|B(x,\delta)|\to1\) [Fremlin, 261D(b)]. Here \(|\cdot|\) is Lebesgue measure, or outer measure for arbitrary sets.

1. A covering lemma

For a norm on \(\mathbb R^2\), a ball is a closed set \(B(x,r)=\{y:\|y-x\|\le r\}\), and \(5B\) is \(B(x,5r)\). We use the Euclidean norm, and the maximum norm, whose balls are the squares \(Q_r(x)=x+[-r,r]^2\).

Lemma 1.1 (five times covering). Let \(\mathcal F\) be a family of balls of one norm with radii at most \(r_0<\infty\). There is a countable disjoint subfamily \(\mathcal G\) such that every ball of \(\mathcal F\) lies in \(5B'\) for some \(B'\in\mathcal G\).

Proof. Let \(\mathcal F_k\) be the balls with radius in \((2^{-k-1}r_0,2^{-k}r_0]\). Choose inductively a maximal disjoint subfamily \(\mathcal G_k\) of the balls of \(\mathcal F_k\) disjoint from all balls in \(\mathcal G_0\cup\dots\cup\mathcal G_{k-1}\); it is countable, since disjoint balls of radius at least \(2^{-k-1}r_0\) contain disjoint balls of rational centers. A ball \(B\in\mathcal F_k\) meets some ball \(B'\) of \(\mathcal G_0\cup\dots\cup\mathcal G_k\), by maximality; the radius of \(B'\) exceeds half the radius of \(B\), so \(B\subseteq5B'\). \(\square\)

2. Monotone functions

Let \(f\) be a bounded function on an open rectangle \(\Omega\subseteq\mathbb R^2\) which is monotone in each variable, nondecreasing or nonincreasing, the kind fixed for each variable. Reflecting a variable, we may assume \(f\) nondecreasing in both. Write \(u\le w\) when \(u_1\le w_1\) and \(u_2\le w_2\); then \(f(u)\le f(w)\) for \(u\le w\). Let \(\|f\|=\sup|f|\).

For \(x\) and \(r>0\) with \(Q_{2r}(x)\subseteq\Omega\) put \[ \begin{aligned} \Delta(x,r)&=f(x+(r,r))-f(x-(r,r))\\ &\ge0 . \end{aligned} \] Every point of \(Q_r(x)\) lies between \(x-(r,r)\) and \(x+(r,r)\), so the oscillation of \(f\) on \(Q_r(x)\) is at most \(\Delta(x,r)\).

Lemma 2.1. For almost every \(x\in\Omega\) there are \(m_x,r_x>0\) with \(\Delta(x,r)\le m_xr\) for \(0<r<r_x\).

Proof. Increments. Put \[ \begin{aligned} H(x,r)&=\int_{x_2-2r}^{x_2+2r}\bigl(f(x_1+2r,y)\\ &\qquad\quad-f(x_1-2r,y)\bigr)\,dy,\\ V(x,r)&=\int_{x_1-2r}^{x_1+2r}\bigl(f(t,x_2+2r)\\ &\qquad\quad-f(t,x_2-2r)\bigr)\,dt . \end{aligned} \] The integrands are monotone in one variable, hence Borel, and nonnegative. For \(u\) in the square \(x+[-2r,-r]^2\) of area \(r^2\), monotonicity gives \(u\le x-(r,r)\) and \(u+(3r,3r)\ge x+(r,r)\), so \[ \begin{aligned} \Delta(x,r)\le{}&\bigl(f(u_1+3r,u_2)\\ &\quad-f(u_1,u_2)\bigr)\\ &+\bigl(f(u_1+3r,u_2+3r)\\ &\quad-f(u_1+3r,u_2)\bigr). \end{aligned} \] In the first bracket \(u_1\ge x_1-2r\) and \(u_1+3r\le x_1+2r\), so it is at most \(f(x_1+2r,u_2)-f(x_1-2r,u_2)\); integrating over \(u\) gives at most \(rH(x,r)\). The second bracket is at most \(f(u_1+3r,x_2+2r)\) minus \(f(u_1+3r,x_2-2r)\), and integrates to at most \(rV(x,r)\). Hence \[ r\Delta(x,r)\le H(x,r)+V(x,r). \]

Disjoint squares. Let \(Q_{2r_i}(x_i)\) be disjoint squares in \(\Omega\). For a fixed \(y\), the horizontal intervals \([x_{i,1}-2r_i,x_{i,1}+2r_i]\) of those squares whose vertical range contains \(y\) are disjoint, so the corresponding increments of the nondecreasing function \(f(\cdot,y)\) add up to at most \(2\|f\|\). Integrating over \(y\) in the bounded vertical range \(h\) of \(\Omega\), \(\sum_iH(x_i,r_i)\le2\|f\|\,h\); similarly \(\sum_iV(x_i,r_i)\le2\|f\|\,w\), with \(w\) the width of \(\Omega\).

Covering. For \(m\ge1\) let \(E_m\) be the set of \(x\in\Omega\) for which \(H(x,r)+V(x,r)>mr^2\) for arbitrarily small \(r\). Applying Lemma 1.1 to the squares \(Q_{2r}(x)\subseteq\Omega\), \(x\in E_m\), with this property and \(r\le1\), gives disjoint squares \(Q_{2r_i}(x_i)\) whose fivefold enlargements cover \(E_m\), so \[ \begin{aligned} |E_m|&\le\sum_i(20r_i)^2\\ &<\frac{400}m\sum_i\bigl(H(x_i,r_i)+V(x_i,r_i)\bigr)\\ &\le\frac{800\|f\|(h+w)}m . \end{aligned} \] The points outside \(\bigcap_mE_m\), a null set, have \(H+V\le m_xr^2\) for all small \(r\), hence \(\Delta(x,r)\le m_xr\). \(\square\)

Theorem 2.2 (Saks; Chabrillac–Crouzeix). A bounded Borel function on an open rectangle in \(\mathbb R^2\) which is monotone in each variable is differentiable at almost every point.

Proof. Assume \(f\) nondecreasing in both variables, as above. For \(m\ge1\) let \(E_m\) be the set of points \(x\) with \(Q_{2/m}(x)\subseteq\Omega\) and \(\Delta(x,r)\le mr\) for every rational \(r\in(0,1/m)\). It is a Borel set, as a countable intersection of conditions on Borel functions of \(x\). By Lemma 2.1, the union of the \(E_m\) contains almost every point of \(\Omega\).

Lipschitz pieces. Let \(x\in E_m\) and \(y\in\Omega\) with \(\rho=\max(|y_1-x_1|,|y_2-x_2|)\). If \(\rho<1/m\), choose rational \(r\in[\rho,1/m)\) close to \(\rho\): \(|f(y)-f(x)|\le\Delta(x,r)\le mr\), so \(|f(y)-f(x)|\le m\rho\). If \(\rho\ge1/m\), \(|f(y)-f(x)|\le2\|f\|\le2\|f\|m\rho\). So the restriction of \(f\) to \(E_m\) is Lipschitz, and it extends to a Lipschitz function \(g_m\) on \(\mathbb R^2\) [Fremlin, 262B(b)], which is differentiable almost everywhere [Fremlin, 262Q].

Squeezing. Let \(x\in E_m\) be a point where \(g_m\) is differentiable and where \(E_m\) has density one [Fremlin, 261D(b)]; almost every point of \(E_m\) is such. Let \(\eta\in(0,\frac12)\). For small \(v\), the square \(y+[-\eta|v|,0]^2\), with \(y=x+v\), lies in \(B(x,2|v|)\) and has area \(\eta^2|v|^2\), a fixed fraction of the area of that ball; by density one it meets \(E_m\) once \(|v|\) is small. So there are points \(y^\pm\in E_m\) with \(y^-\le y\le y^+\) and \(|y^\pm-y|\le\sqrt2\,\eta|v|\). By monotonicity and \(f=g_m\) on \(E_m\), \[ \begin{aligned} &g_m(y^-)\le f(y)\le g_m(y^+),\\ &g_m(y^\pm)=g_m(x)\\ &\qquad+Dg_m(x)(y^\pm-x)+o(|v|), \end{aligned} \] so \[ \begin{aligned} &|f(x+v)-f(x)-Dg_m(x)v|\\ &\quad\le\sqrt2\,\eta\,\|Dg_m(x)\|\,|v|+o(|v|). \end{aligned} \] Since \(\eta\) is arbitrary, \(f\) is differentiable at \(x\) with derivative \(Dg_m(x)\). \(\square\)

3. Critical images

Lemma 3.1 (critical images). Let \(\Omega\subseteq\mathbb R^2\) be open and \(F:\Omega\to\mathbb R^2\) any map. Let \(S\) be the set of points where \(F\) is differentiable with \(\operatorname{rank}DF(x)<2\). Then \(F(S)\) has outer measure zero, so it lies in a Borel null set.

Proof. Fix \(\rho,m\ge1\) and let \(S'\) be the set of \(x\in S\) with \(|x|<\rho\) and \(\|DF(x)\|\le m\). Fix \(\epsilon\in(0,1)\). For each \(x\in S'\) there are arbitrarily small \(r<1\) with \(B(x,5r)\subseteq\Omega\) and \[ \begin{aligned} &|F(y)-F(x)-DF(x)(y-x)|\\ &\quad\le\epsilon|y-x| \end{aligned} \] for \(y\in B(x,5r)\). Lemma 1.1, for Euclidean balls, gives disjoint balls \(B(x_i,r_i)\) of this kind whose fivefold enlargements cover \(S'\). They lie in \(B(0,\rho+1)\), so \(\sum_i\pi r_i^2\le\pi(\rho+1)^2\). Since \(DF(x_i)\) has rank at most one, \(DF(x_i)(B(0,5r_i))\) lies in a segment of length \(10mr_i\), and \(F(B(x_i,5r_i))\) lies in the \(5\epsilon r_i\)-neighbourhood of a translate of it, whose area is at most \((100m\epsilon+25\pi\epsilon^2)r_i^2\). So the outer measure of \(F(S')\) is at most \((100m\epsilon+25\pi\epsilon^2)(\rho+1)^2\), for every \(\epsilon\). Take the union over \(\rho\) and \(m\). \(\square\)

This is the easy part of Sard's theorem, but without any continuity of \(F\) away from the point: only differentiability at the points of \(S\) is used. In the course, \(F\) is a pair of limits of rank coordinates, known to be differentiable almost everywhere but not continuous.

4. Exercises

4.1 (easy). Show that \(f(x,y)=\mathbf 1_{x+y>0}\) is monotone in each variable, find where it is not differentiable, and check that this set is null.

4.2 (medium). Show that Theorem 2.2 fails for functions monotone in one variable only: give a bounded Borel function on the square, nondecreasing in \(x\), that is differentiable nowhere.

4.3 (easy). For \(F(x,y)=(x,x^2)\), describe \(S\) and \(F(S)\) in Lemma 3.1.

5. Solutions

4.1. Increasing \(x\) or \(y\) can only turn \(x+y>0\) from false to true. Off the line \(x+y=0\) the function is locally constant, hence differentiable; on the line it jumps. The line is null.

4.2. Let \(f(x,y)=x+\mathbf 1_{y\in\mathbb Q}\). For each fixed \(y\) it is increasing in \(x\), but at every point it is discontinuous in \(y\), hence not differentiable.

4.3. \(DF=\begin{pmatrix}1&0\\2x&0\end{pmatrix}\) has rank one everywhere, so \(S=\mathbb R^2\), and \(F(S)\) is the parabola \(\{(x,x^2)\}\), a null set.

References