Bounded Twice Continuously Differentiable Functions with Bounded First and Second 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 classes $C^{1}$ and $C^{2}$ on $\mathbb{R}^{q}$, the partial derivatives $\partial_{i}\psi$ and the iterated partial derivatives $\partial_{j}\partial_{i}\psi$ 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 $C^{2}$ functions)¶ $C^{2}_{b}(\mathbb{R}^{q})$ is the set of the functions $\psi:\mathbb{R}^{q}\to\mathbb{R}$ of class $C^{2}$ on $\mathbb{R}^{q}$ such that $\psi$, $\partial_{i}\psi$ and $\partial_{j}\partial_{i}\psi$ are bounded for all $i,j\in[q]$; for such $\psi$ these partial derivatives exist at every point of $\mathbb{R}^{q}$ by clauses 1, 2 and 4 of C^k Maps on a Euclidean Open Set, read through its clause 3.