A Confining Potential and Its Derivative are Continuous and Borel

The real line is identified with $\mathbb{R}^{1}$ as in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative; a function $\mathbb{R}\to\mathbb{R}$ is continuous if it is continuous on $\mathbb{R}$ for the absolute-value metric, and Borel is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. Let $V$ be a confining potential, with derivative $V'$.

1. (Continuity)¶ The functions $V$ and $V'$ are continuous.

2. (Measurability)¶ The functions $V$ and $V'$ are Borel.