Mollification at a Point of Twice Differentiability: Convergence of the Mollified Gradient and Hessian, and a Hessian Bound for Lipschitz Functions

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 $f:\mathbb{R}^{n}\to\mathbb{R}$ be continuous on $\mathbb{R}^{n}$, let $\delta\in\mathbb{R}$ with $0<\delta$, and let $\rho$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$. For $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ let $\rho_{\varepsilon}(y)=(\varepsilon^{-1})^{n}\rho(\varepsilon^{-1}y)$ for $y\in\mathbb{R}^{n}$, a mollifier kernel of radius $\varepsilon\delta$ by Rescaling a Mollifier Kernel, and let $f_{\varepsilon}=f*\rho_{\varepsilon}$ be the convolution, taken with $\Omega=\mathbb{R}^{n}$, which is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous; it is defined on all of $\mathbb{R}^{n}$, every closed ball of $(\mathbb{R}^{n},d_{E})$ being contained in $\mathbb{R}^{n}$, and it is smooth on $\mathbb{R}^{n}$ by claim 2 of Convolution with a $C^k$ Kernel is of Class $C^k$. Its gradient $Df_{\varepsilon}(x)$ and Hessian matrix $D^{2}f_{\varepsilon}(x)\in\mathcal{S}(n)$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives.

1. (Convergence at a point of twice differentiability)¶ Let $x,p\in\mathbb{R}^{n}$ and $B\in\mathcal{S}(n)$ be such that $f$ is twice differentiable at $x$ with first-order coefficient $p$ and Hessian $B$, and let $(\varepsilon_{m})_{m\in\mathbb{N}}$ be a sequence of positive real numbers converging to $0$. Then $(Df_{\varepsilon_{m}}(x))_{m\in\mathbb{N}}$ converges to $p$ in $\mathbb{R}^{n}$, and $(D^{2}f_{\varepsilon_{m}}(x))_{m\in\mathbb{N}}$ converges to $B$ in $\mathcal{S}(n)$.

2. (A Hessian bound for Lipschitz functions)¶ There is a nonnegative real number $K$, depending only on $n$, $\delta$ and $\rho$, with the following property. Let $L\in\mathbb{R}$ with $0\le L$, and suppose that $f$ is Lipschitz with constant $L$ from $(\mathbb{R}^{n},d_{E})$ to $\mathbb{R}$ with the absolute-value metric. Then for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ and every $x\in\mathbb{R}^{n}$,