Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space

Adopt Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation and fix a dimension $q$. The set $\mathbb{R}^{q}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous; the class $C^{1}$ on $\mathbb{R}^{q}$ and the partial derivatives $\partial_{i}\psi$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, and a real-valued function is bounded as defined there.

(Bounded $C^{1}$ functions)¶ $C^{1}_{b}(\mathbb{R}^{q})$ is the set of the functions $\psi:\mathbb{R}^{q}\to\mathbb{R}$ of class $C^{1}$ on $\mathbb{R}^{q}$ such that $\psi$ and $\partial_{i}\psi$ are bounded for every $i\in[q]$; for such $\psi$ the partial derivative $\partial_{i}\psi(x)$ exists at every $x\in\mathbb{R}^{q}$ by clause 1 of C^k Maps on a Euclidean Open Set, read through its clause 3.