Local Lipschitz Bound and Continuity for a Semiconvex Function on an Open Convex Set

Let $n$ be a natural number with $1\le n$ and let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure and the absolute value $|\cdot|$. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points, the scalar multiple, and the difference $y-x$ of points; write $\lVert\,\cdot\,\rVert$ for the Euclidean norm and $d_{E}$ for the Euclidean distance, a metric on $\mathbb{R}^{n}$ satisfying $d_{E}(x,y)=\lVert x-y\rVert$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Closed balls $\bar{B}_{d_{E}}(x,\rho)$ are those of Closed Ball in a Metric Space.

Let $U\subseteq\mathbb{R}^{n}$ be convex and open in $(\mathbb{R}^{n},d_{E})$, let $\mu\in\mathbb{R}$ satisfy $0\le\mu$, and let $\varphi:U\to\mathbb{R}$ be semiconvex on $U$ with constant $\mu$. Then the following hold.

1. (Local Lipschitz bound) (i) For every $x_{0}\in U$ there are $\rho,L\in\mathbb{R}$ with $0<\rho$ and $0\le L$ such that $\bar{B}_{d_{E}}(x_{0},\rho)\subseteq U$ and

$$\bigl|\varphi(y)-\varphi(x)\bigr|\le L\,\lVert y-x\rVert\qquad\text{for all }x,y\in\bar{B}_{d_{E}}(x_{0},\rho).$$

2. (Continuity) (ii) For every $S\subseteq U$, every $x\in S$ and every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$, there is $\delta\in\mathbb{R}$ with $0<\delta$ such that every $y\in S$ satisfying $\lVert y-x\rVert<\delta$ satisfies

$$\bigl|\varphi(y)-\varphi(x)\bigr|<\varepsilon .$$