Let $u \in BV(\mathbb{R}^n)$ and $\varepsilon > 0$. Then there exists $g \in C^1(\mathbb{R}^n)$ such that \begin{align*} \|u - g\|_{L^1(\mathbb{R}^n)} < \varepsilon \quad \text{and} \quad \Bigl| |Du|(\mathbb{R}^n) - \int_{\mathbb{R}^n} |\nabla g| \, d\mathcal{L}^n \Bigr| < \varepsilon. \end{align*}

Let $1 \leq p < \infty$, and let $u \in W^{1,p}(\mathbb{R}^n)$ and $\varepsilon > 0$. Then there exists $g \in C^1(\mathbb{R}^n)$ such that \begin{align*} \mathcal{L}^n(\{u \neq g\}) < \varepsilon \quad \text{and} \quad \|u - g\|_{W^{1,p}(\mathbb{R}^n)} < \varepsilon. \end{align*}

Let $f: \mathbb{R}^n \to \mathbb{R}$ be Lipschitz and let $\varepsilon > 0$. Then there exists $g \in C^1(\mathbb{R}^n)$ such that \begin{align*} \mathcal{L}^n\bigl(\{x \in \mathbb{R}^n : f(x) \neq g(x)\} \cup \{x \in \mathbb{R}^n : \nabla f(x) \neq \nabla g(x)\}\bigr) < \varepsilon. \end{align*} Moreover, $g$ can be chosen Lipschitz with $\operatorname{Lip}(g) \leq C\, \operatorname{Lip}(f)$ for a dimensional constant $C$.

Let $K \subseteq \mathbb{R}^n$ be closed and nonempty, and let $f, d : K \to \mathbb{R}$ be continuous functions satisfying the Whitney $C^1$ condition. Then there exists $\tilde{f} \in C^1(\mathbb{R}^n)$ such that \begin{align*} \tilde{f}|_K = f \quad \text{and} \quad \nabla \tilde{f}|_K = d. \end{align*} Moreover, if $f$ and $d$ are bounded, then $\tilde{f}$ can be chosen so that $\|\nabla \tilde{f}\|_{L^\infty(\mathbb{R}^n)}$ is controlled by the data.

Let $f: U \to \mathbb{R}$ be convex on the open convex set $U \subseteq \mathbb{R}^n$. Then the Monge–Ampère measure $\mu_f$ is absolutely continuous with respect to $\mathcal{L}^n$ on the set where $f$ is twice differentiable, and \begin{align*} d\mu_f = \det D^2 f \, d\mathcal{L}^n \end{align*} on the set $\Omega_2 := \{x \in U: f \text{ is twice differentiable at } x\}$, which has full $\mathcal{L}^n$-measure in $U$. In full, for every Borel set $E \subseteq U$, \begin{align*} \mu_f(E) = \int_{E \cap \Omega_2} \det D^2 f(x)\, d\mathcal{L}^n(x). \end{align*}

Let $U \subseteq \mathbb{R}^n$ be open and convex, and let $f: U \to \mathbb{R}$ be convex. Then $f$ is twice differentiable $\mathcal{L}^n$-almost everywhere on $U$: for $\mathcal{L}^n$-a.e. $x_0 \in U$ there exists a symmetric matrix $A(x_0) \in \mathbb{R}^{n \times n}$ such that \begin{align*} f(x) = f(x_0) + \nabla f(x_0) \cdot (x - x_0) + \tfrac{1}{2}(x - x_0)^\top A(x_0)(x - x_0) + o(|x - x_0|^2) \end{align*} as $x \to x_0$. The matrix $A(x_0)$ is the pointwise Hessian of $f$ at $x_0$, denoted $D^2 f(x_0)$, and satisfies $D^2 f(x_0) \ge 0$ (positive semidefinite) at every point of twice differentiability.

Let $f: U \to \mathbb{R}$ be convex on the open convex set $U \subseteq \mathbb{R}^n$. Then: (i) For every $x \in U$, $\partial f(x)$ is nonempty, compact, and convex. (ii) **(Monotonicity)** For any $x, y \in U$, $p \in \partial f(x)$, $q \in \partial f(y)$, \begin{align*} (p - q) \cdot (x - y) \ge 0. \end{align*} (iii) At every point $x$ where $f$ is differentiable, $\partial f(x) = \{\nabla f(x)\}$. (iv) The graph of $\partial f$, defined as $\{(x, p) : x \in U,\, p \in \partial f(x)\}$, is a closed subset of $U \times \mathbb{R}^n$.

Let $U \subseteq \mathbb{R}^n$ be open and convex, and let $f: U \to \mathbb{R}$ be convex. Then $f$ is locally Lipschitz on $U$: for every compact set $K \subseteq U$, there exists a constant $L$ depending only on $K$, $f$, and the distance from $K$ to $\partial U$, such that \begin{align*} |f(x) - f(y)| \le L|x - y| \qquad \text{for all } x, y \in K. \end{align*}

Let $1 \leq p \leq n$ and $E \subseteq \mathbb{R}^n$. If $\mathrm{Cap}_p(E) = 0$, then $\mathcal{H}^s(E) = 0$ for every $s > n - p$.

Let $\Omega \subseteq \mathbb{R}^n$ be open, $p > n$, and $u \in W^{1,p}_{\mathrm{loc}}(\Omega)$. Then $u$ is classically differentiable at $\mathcal{L}^n$-a.e. point of $\Omega$: for $\mathcal{L}^n$-a.e. $x_0 \in \Omega$, there exists a vector $\ell \in \mathbb{R}^n$ such that \begin{align*} \lim_{r \to 0} \frac{|u(x_0 + r h) - u(x_0) - r\, \ell \cdot h|}{r} = 0 \end{align*} uniformly in $h$ with $|h| = 1$. Moreover, $\ell = \nabla u(x_0)$, the weak gradient evaluated at $x_0$.