Proof of A Confining Potential and Its Derivative are Continuous and Borel

Each result cited is universally quantified over the data in its own statement. Let $f$ be either of the functions $V$ and $V'$.

Claim 1. By claim 1 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line, $\mathbb{R}$ is an interval and every $x\in\mathbb{R}$ is an interior point of it. By Confining Potentials on the Real Line §confining, $V$ is differentiable at every point of $\mathbb{R}$ with derivative $V'$, and $V'$ is differentiable at every point of $\mathbb{R}$; so $f$ is differentiable at every $x\in\mathbb{R}$. Fix $x\in\mathbb{R}$. By Differentiability at an Interior Point Implies Continuity There, applied with the interval $I=\mathbb{R}$, the function $f$ and the interior point $x_{0}=x$, the function $f$ is continuous at $x$ relative to $\mathbb{R}$, as a map from $\mathbb{R}$ into $\mathbb{R}$ with the absolute-value metric $d_{\mathbb{R}}$ on both sides. As $x$ was arbitrary, $f$ is continuous on $\mathbb{R}$, that is, continuous.

Claim 2. By The Euclidean Distance on the Real Line is the Absolute Value Metric, the Euclidean distance of $\mathbb{R}^{1}$ is the absolute-value metric $d_{\mathbb{R}}$. By claim 1, $f$ is therefore continuous for the Euclidean distance of $\mathbb{R}^{1}$ and the absolute-value metric of $\mathbb{R}$, and hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps.