$C^k$ Maps on a Euclidean Open Set

Let $n,m$ be natural numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$, let $\mathbb{R}$ be the real numbers, and let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$, with coordinate functions $f_j:U\to\mathbb{R}$. For a natural number $k$, we define what it means for $f$ to be of class $C^k$ on $U$, recursively in $k$.

1. (Class $C^1$) $f$ is of class $C^1$ on $U$ if, for every $j\in\{1,\dots,m\}$, the coordinate function $f_j$ is continuous at every point of $U$ and, for every $i\in\{1,\dots,n\}$, the partial derivative of $f_j$ with respect to the $i$th variable exists at every point of $U$ and the function $\partial_i f_j:U\to\mathbb{R}$, $x\mapsto\partial_i f_j(x)$, is continuous at every point of $U$.

2. (Class $C^{k+1}$) For a natural number $k$, $f$ is of class $C^{k+1}$ on $U$ if $f$ is of class $C^1$ on $U$ and, for every $j\in\{1,\dots,m\}$ and every $i\in\{1,\dots,n\}$, the function $\partial_i f_j:U\to\mathbb{R}$ is of class $C^k$ on $U$ (in the sense of clause 3).

3. (Scalar convention) A function $g:U\to\mathbb{R}$ is of class $C^k$ on $U$ if it is of class $C^k$ as a map into $\mathbb{R}^1$ with single coordinate function $g$.

4. (Iterated partial derivatives) For $i,l\in\{1,\dots,n\}$ and $j\in\{1,\dots,m\}$ such that the partial derivative of $f_j$ with respect to the $i$th variable exists at every point of $U$ and the resulting function $\partial_i f_j:U\to\mathbb{R}$ has a partial derivative with respect to the $l$th variable at every point of $U$, we write $\partial_l\partial_i f_j$ for the function from $U$ to $\mathbb{R}$ whose value at $x\in U$ is that partial derivative at $x$; for a function $g:U\to\mathbb{R}$ regarded as in clause 3, whose single coordinate function is $g$ itself, this notation reads $\partial_l\partial_i g$.