A Continuous Lattice-Periodic Function is Uniformly Continuous

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background and in the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, both used here with a natural number $n$ satisfying $1\le n$, exactly as in Lattice-Periodic Functions and the Periodic Function Classes, whose reading of continuity on $\mathbb{R}^{n}$ is in force. From the first setting we take Euclidean space $\mathbb{R}^{n}$ with its norm $\lVert\,\cdot\,\rVert$, distance $d_{E}$, topology and the notions of closed, bounded and compact set. Let $\mathbb{Z}^{n}$ be the integer lattice, let $\mathbb{Z}^{n}$-periodicity be as defined there, and let $C_{\mathrm{per}}$ abbreviate the class $C_{\mathrm{per}}(\mathbb{R}^{n})$.

1. (Uniform continuity)(i) Let $w\in C_{\mathrm{per}}$. Then $w$ is uniformly continuous as a map from $\mathbb{R}^{n}$ with $d_{E}$ to $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Numbers. Explicitly, for every real number $\theta$ with $0<\theta$ there is a real number $r$ with $0<r$ such that

$$|w(x)-w(z)|\le\theta\qquad\text{for all }x,z\in\mathbb{R}^{n}\text{ with }\lVert x-z\rVert\le r .$$