This course builds on the measure-theoretic foundations of Geometric Measure Theory I to develop the machinery of differentiation for non-smooth functions. The central focus is on Lipschitz maps, functions with bounded distributional derivatives, and convex functions—exotic in classical analysis but ubiquitous in geometric variational problems and optimal transport. Rather than assume differentiability, we will establish when and how non-smooth maps admit meaningful notions of derivative, culminating in two transformative theorems: the area formula and coarea formula. These generalizations of the classical change-of-variables rule extend Fubini's theorem and integration by parts to all Lipschitz maps, unlocking geometric measure theory's power to handle sets and functions of arbitrary regularity.

The narrative arc begins with Rademacher's theorem, which asserts that every Lipschitz function is almost everywhere differentiable—turning even the roughest Lipschitz maps into objects of differential geometry. We then develop the Jacobian for such maps, culminating in the area formula: a measure-theoretic identity that counts how the pullback of a Lipschitz map distributes mass across fibers of its image. Inverting this perspective yields the coarea formula, which slices a domain along level sets of a Lipschitz function and integrates the resulting $(n-1)$-dimensional areas. Beyond these formulas, we investigate finer differentiability results for Sobolev functions, discover Alexandrov's theorem on the twice-differentiability of convex functions, and prove Whitney's extension theorem—the technical backbone that underpins $C^1$ approximation of rough functions.

Each chapter deepens the toolkit: approximate differentiability prepares the ground for Sobolev regularity; linear maps and Jacobians establish a finite-dimensional calculus; the area and coarea formulas unify geometric and analytic perspectives; Alexandrov and Whitney theory extend differentiability to broader classes. The course culminates in worked examples demonstrating how these abstract results illuminate concrete geometry—from minimal surfaces to regularity of solutions to elliptic PDEs. Throughout, the emphasis is on transforming regularity assumptions into measure-theoretic identities, enabling us to compute and reason about singular sets that classical analysis cannot reach.

# 1. Lipschitz Functions and Rademacher's Theorem

This chapter lays the analytic foundation for everything that follows in the course. The central object is the class of Lipschitz functions, which are maps controlled by a linear modulus of continuity. The central theorem — Rademacher's theorem — asserts that any locally Lipschitz function $f : \mathbb{R}^n \to \mathbb{R}^m$ is differentiable at Lebesgue-almost every point. This is a remarkable fact: differentiability is a delicate analytic condition, while the Lipschitz condition is purely metric, yet the metric condition forces almost-everywhere differentiability. Every result in this course — the area formula, the coarea formula, and the finer differentiability theorems — rests on this foundation.

## Lipschitz Functions: Definitions and Basic Properties

[motivation] ### Why Lipschitz? The classical change-of-variables formula for $C^1$ maps requires genuine everywhere differentiability. Yet many natural maps arising in geometry and analysis — distance functions, convex functions, solutions to certain PDEs — are not $C^1$. They may fail to be differentiable at corners, kinks, or on measure-zero sets. The Lipschitz condition is the minimal quantitative regularity needed to control how a map distorts measure. A function with Lipschitz constant $L$ expands distances by at most $L$, which bounds by how much it can stretch $k$-dimensional volume. This makes Lipschitz maps the natural domain for GMT, and Rademacher's theorem is the theorem that upgrades "metric control" to "almost-everywhere differential calculus." ### The Lipschitz Condition Versus Continuity A function can be uniformly continuous without being Lipschitz — consider $f(x) = \sqrt{x}$ on $[0,1]$, which has modulus of continuity $\omega(\delta) = \sqrt{\delta}$ but is not Lipschitz because $|f(x) - f(0)|/|x - 0| = x^{-1/2} \to \infty$ as $x \to 0^+$. The Lipschitz condition imposes a linear modulus and is therefore strictly between uniform continuity and $C^1$. For GMT the linear modulus is essential: it is precisely the hypothesis under which areas and coareas can be computed by integrating the Jacobian. [/motivation]

The core definition precisely quantifies what it means to control a function's rate of oscillation.

[definition: Lipschitz Function] Let $E \subseteq \mathbb{R}^n$ and $m \ge 1$. A function $f : E \to \mathbb{R}^m$ is **Lipschitz** (or **Lipschitz continuous**) if there exists a constant $L \ge 0$ such that \begin{align*} |f(x) - f(y)| \le L|x - y| \quad \text{for all } x, y \in E. \end{align*} The **Lipschitz constant** of $f$, denoted $\operatorname{Lip}(f)$, is the infimum of all such $L$: \begin{align*} \operatorname{Lip}(f) := \sup_{\substack{x, y \in E \\ x \neq y}} \frac{|f(x) - f(y)|}{|x - y|}. \end{align*} We say $f$ is **locally Lipschitz** if every point $x \in E$ has a neighbourhood $U$ such that $f|_{U \cap E}$ is Lipschitz. [/definition]

When $E$ is open and $f$ is locally Lipschitz, the Lipschitz constant may vary from region to region. For compact subsets, however, a locally Lipschitz function is automatically globally Lipschitz with some constant depending on the compact set.

[example: Standard Examples of Lipschitz Functions] **Distance function.** Fix a non-empty set $A \subseteq \mathbb{R}^n$ and define $d_A(x) = \operatorname{dist}(x, A) = \inf_{a \in A} |x - a|$. For any $x, y \in \mathbb{R}^n$ and any $a \in A$, the triangle inequality gives $d_A(x) \le |x - a| \le |x - y| + |y - a|$, so taking the infimum over $a$ yields $d_A(x) \le |x - y| + d_A(y)$. Exchanging $x$ and $y$ gives $|d_A(x) - d_A(y)| \le |x - y|$, so $d_A$ is $1$-Lipschitz, i.e., $\operatorname{Lip}(d_A) = 1$. **Linear maps.** Any linear map $T : \mathbb{R}^n \to \mathbb{R}^m$ is Lipschitz with $\operatorname{Lip}(T) = \|T\|_{\mathcal{L}(\mathbb{R}^n, \mathbb{R}^m)}$, the operator norm. This follows immediately from $|T(x) - T(y)| = |T(x-y)| \le \|T\| |x - y|$. **Absolute value.** The function $f(x) = |x|$ on $\mathbb{R}$ satisfies $||x| - |y|| \le |x - y|$ by the reverse triangle inequality, so $\operatorname{Lip}(f) = 1$. Note that $f$ is not differentiable at $x = 0$. **Smooth functions on compact sets.** If $f \in C^1(\overline{U})$ for a bounded open set $U$, then $\operatorname{Lip}(f) \le \sup_{x \in \overline{U}} |\nabla f(x)|$ by the mean value theorem. This shows $C^1(\overline{U}) \subseteq \operatorname{Lip}(\overline{U})$, but the inclusion is strict: the function $f(x) = |x|$ on $\overline{B}(0,1) \subset \mathbb{R}$ is Lipschitz but fails to be $C^1$ at $x = 0$. [/example]

Having established the central examples, we record the closure properties that make the class of Lipschitz functions easy to work with in practice.

[remark: Closure Properties of Lipschitz Functions] The class of Lipschitz functions on $E \subseteq \mathbb{R}^n$ is closed under the following operations, with explicit Lipschitz constants: - **Sums:** $\operatorname{Lip}(f + g) \le \operatorname{Lip}(f) + \operatorname{Lip}(g)$. - **Products:** If $f, g$ are Lipschitz and bounded, then $fg$ is Lipschitz with $\operatorname{Lip}(fg) \le \|f\|_\infty \operatorname{Lip}(g) + \|g\|_\infty \operatorname{Lip}(f)$. - **Compositions:** $\operatorname{Lip}(f \circ g) \le \operatorname{Lip}(f) \cdot \operatorname{Lip}(g)$. - **Pointwise max and min:** $\operatorname{Lip}(\max(f,g)) \le \max(\operatorname{Lip}(f), \operatorname{Lip}(g))$, and similarly for $\min$. - **Uniform limits:** If $f_k \to f$ uniformly and each $f_k$ is $L$-Lipschitz, then $f$ is $L$-Lipschitz. [/remark]

The closure under uniform limits is particularly useful: it allows Lipschitz functions to be constructed as uniform limits of smooth approximations without increasing the Lipschitz constant. This connects to mollification arguments that appear throughout the proofs in later chapters.

### Absolute Continuity on Lines

One of the most important structural properties of Lipschitz functions is their behaviour along lines. For a function $f : \mathbb{R}^n \to \mathbb{R}$ that is $L$-Lipschitz, the restriction to any line $\ell$ is $L$-Lipschitz as a function of the arc-length parameter. A one-variable $L$-Lipschitz function on $\mathbb{R}$ is absolutely continuous, and an absolutely continuous function is differentiable almost everywhere (with respect to one-dimensional Lebesgue measure $\mathcal{L}^1$). This one-dimensional observation is the seed from which Rademacher's theorem grows.

[definition: Absolutely Continuous on Lines] A function $f : U \to \mathbb{R}$ defined on an open set $U \subseteq \mathbb{R}^n$ is **absolutely continuous on lines (ACL)** if for $\mathcal{L}^{n-1}$-almost every line $\ell$ parallel to each coordinate axis, the restriction $f|_\ell$ is absolutely continuous on each compact subinterval of $\ell \cap U$. [/definition]

Every locally Lipschitz function is ACL, because the Lipschitz condition on the ambient space restricts to a Lipschitz (hence absolutely continuous) condition on each line. The ACL property is important for Sobolev theory as well — it characterises $W^{1,p}_{\mathrm{loc}}(U)$ for $p \ge 1$ via Nikodym's theorem — but for our purposes, what matters is that ACL implies the existence of partial derivatives almost everywhere, which Rademacher upgrades to full differentiability almost everywhere.

## Extension Theorems: Kirszbraun and McShane

A priori, a Lipschitz function is defined only on a subset $E$ of $\mathbb{R}^n$. When the proof of Rademacher's theorem reduces to the scalar case, one needs to work with a function on all of $\mathbb{R}^n$, not merely on $E$. The naive attempt — extend $f$ by zero outside $E$ — immediately fails: if $E = [1, 2]$ and $f \equiv 1$, then the zero extension has a jump discontinuity at the boundary and is nowhere near Lipschitz across it. Can one always extend while preserving the Lipschitz constant? For real-valued functions the answer is yes, and the construction is explicit.