Subgradients near a Point of Twice Differentiability of a Convex Function, and Invariance of the Second-Order Expansion under Lipschitz Truncation

In the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, used with a natural number $n$ satisfying $1\le n$, let $U\subseteq\mathbb{R}^{n}$ be open and convex, let $f:U\to\mathbb{R}$ be convex on $U$, with subdifferential $\partial_{U}f$, and let $y\in U$, $p\in\mathbb{R}^{n}$ and $B\in\mathcal{S}(n)$ be such that $f$ is twice differentiable at $y$ with first-order coefficient $p$ and Hessian $B$. Then the following hold.

1. (A single subgradient)¶ $\partial_{U}f(y)=\{p\}$.

2. (First-order expansion of the subgradients)¶ For every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\delta\in\mathbb{R}$ with $0<\delta$ such that every $y'\in\mathbb{R}^{n}$ with $\lVert y'-y\rVert<\delta$ lies in $U$ and every $q\in\partial_{U}f(y')$ satisfies

3. (Lipschitz truncation)¶ Let $L\in\mathbb{R}$ satisfy $\lVert p\rVert<L$, and let $f^{L}:\mathbb{R}^{n}\to\mathbb{R}$ be the Lipschitz truncation of $f$ at level $L$, which is defined because $p\in\partial_{U}f(y)$ by claim 1. Then there is $r\in\mathbb{R}$ with $0<r$ such that every $y'\in\mathbb{R}^{n}$ with $\lVert y'-y\rVert<r$ lies in $U$ and satisfies $f^{L}(y')=f(y')$, and $f^{L}$ is twice differentiable at $y$ with first-order coefficient $p$ and Hessian $B$.