The Classes Cᵏ(U) Exist and Are Unique

In the setting of Higher Derivatives on Open Subsets of ℝⁿ: Notation, let $\mathbb{R}^{U}$ be the set of functions $U\to\mathbb{R}$ (a set by The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (ii)) and $\mathcal{P}(\mathbb{R}^{U})$ its power set.

(i) Existence and uniqueness. There is exactly one function $k\mapsto\mathcal{C}^{k}(U)$ from $\mathbb{N}_{0}$ to $\mathcal{P}(\mathbb{R}^{U})$ such that

$$\mathcal{C}^{0}(U)=\{h\in\mathbb{R}^{U}:\ h\text{ is continuous on }U\}$$

and, for every $k\in\mathbb{N}_{0}$,

$$\mathcal{C}^{k+1}(U)=\bigl\{h\in\mathbb{R}^{U}:\ \text{for every }i\in[n],\ \partial_{i}h(x)\text{ exists for every }x\in U\text{ and }\partial_{i}h\in\mathcal{C}^{k}(U)\bigr\},$$

where continuity on $U$ is that of Continuous Map Between Metric Spaces (ii).