$C^k$ maps and multi-index derivative notation in Euclidean space

Let $m\ge1$, let $U\subseteq\mathbb R^m$ be open, and let $f:U\to\mathbb R$. A multi-index is $\alpha=(\alpha_0,\ldots,\alpha_{m-1})\in\mathbb N^m$. Put

$|\alpha|:=\sum_{i<m}\alpha_i,\qquad \alpha!:=\prod_{i<m}\alpha_i!,\qquad h^\alpha:=\prod_{i<m}h_i^{\alpha_i}\quad(h\in\mathbb R^m).$

Here $|\alpha|$ and $\alpha!$ use the natural-number sum and product of Finite sums and finite products of natural numbers, $\sum_{k<n} a_k$ and $\prod_{k<n} a_k$ in $\mathbb{N}$, and $n!$ is the factorial of The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$. By contrast, $h^\alpha$ is the finite product in $\mathbb R$ of Finite sums and finite products, by recursion, with the natural exponents interpreted by Integer powers $a^m$. For the zero multi-index $0$, set $D^0f:=f$. For nonzero $\alpha$, write

$D^\alpha f:=\partial_0^{\alpha_0}\cdots\partial_{m-1}^{\alpha_{m-1}}f$

for this displayed, canonical order whenever it exists. Coordinate partial derivatives have the meaning fixed in Directional derivatives and partial derivatives of a map $U\subseteq\mathbb{R}^m\to\mathbb{R}^n$.

For $k\in\mathbb N$, $f$ is of class $C^k$ on $U$ when, for every word $(i_1,\ldots,i_r)$ of coordinate indices with $0\le r\le k$, the iterated derivative $\partial_{i_r}\cdots\partial_{i_1}f$ exists and is continuous on $U$; the word of length $0$ denotes $f$. Thus this definition does not presuppose that differently ordered derivatives are equal. Equality of their values is a later theorem under these regularity hypotheses.