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

Let $m\ge 1$, let $U\subseteq {\mathbb{R}}^m$ be open, and let $f:U\to \mathbb{R}$. A multi-index is $\alpha =({\alpha }_0,\dots ,{\alpha }_{m-1})\in {\mathbb{N}}^m$. Put

$\mathrm{\mid }\alpha \mathrm{\mid }:={\sum }_{i<m}{\alpha }_i,\alpha !:={\prod }_{i<m}{\alpha }_i!,h^{\alpha }:={\prod }_{i<m}{h}_i^{{\alpha }_i}(h\in {\mathbb{R}}^m)\mathrm{.}$

Here $\mathrm{\mid }\alpha \mathrm{\mid }$ 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 $n^{\underset{‾}{k}}$, 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 $D^{0}f:=f$. For nonzero $\alpha$, write

$D^{\alpha }f:={\mathrm{\partial }}_0^{{\alpha }_0}\cdots {\mathrm{\partial }}_{m-1}^{{\alpha }_{m-1}}f$

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 $(i_1,\dots ,i_r)$ of coordinate indices with $0\le r\le k$, the iterated derivative ${\mathrm{\partial }}_{i_r}\cdots {\mathrm{\partial }}_{i_1}f$ 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.