Proof of Multi-Index Partial Derivatives of a $C^k$ Map and of a Smooth Map

Throughout, "clause 1" and "clause 2" of C^k Maps on a Euclidean Open Set and "clause 1" and "clause 2" of Partial Derivative of a Multi-Index on a Euclidean Open Set are named explicitly when used, and we use the scalar convention of clause 3 of C^k Maps on a Euclidean Open Set without further comment.

Two preliminary observations.

(a) Largest nonzero index. Let $\beta$ be a multi-index of length $n$ with $\beta\ne 0$. Then there is an index $i$ with $1\le i\le n$, $1\le\beta_i$, and $\beta_l=0$ for every $l$ with $i<l\le n$. Indeed, let $T$ be the set of natural numbers $j$ with $1\le j\le n$ such that $\beta_l=0$ for every $l$ with $j<l\le n$. Then $n\in T$, so $T$ is nonempty and has a least element $i$, which satisfies $\beta_l=0$ for every $l$ with $i<l\le n$. If $\beta_i=0$, then $\beta_l=0$ for every $l$ with $i\le l\le n$; in that case, if $2\le i$ then $i-1\in T$, contradicting the minimality of $i$, while if $i=1$ then every entry of $\beta$ vanishes, contradicting $\beta\ne 0$. Hence $1\le\beta_i$.

(b) Order of a difference. Since the order of a multi-index is the sum of its entries, and $\beta-e_i$ differs from $\beta$ only in the $i$th entry, which is smaller by one, we have $|\beta|=|\beta-e_i|+1$ whenever $1\le\beta_i$. In particular a multi-index $\beta$ with $\beta\ne 0$ has $1\le|\beta|$, so its order is a natural number; and if $|\beta|=1$ then $\beta=e_i$ for the index $i$ of observation (a), since the entries are nonnegative and sum to $1$.

Claim 1. By claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, $F$ of class $C^{k}$ on $U$ has every coordinate function $F_j$ of class $C^{k}$ on $U$, and by the final paragraph of Partial Derivative of a Multi-Index on a Euclidean Open Set the map $\partial^{\alpha}F$ exists on $U$ as soon as $\partial^{\alpha}F_j$ exists on $U$ for every $j$ with $1\le j\le m$. It therefore suffices to prove, for every natural number $k$, the statement $Q(k)$: for every $g:U\to\mathbb{R}$ of class $C^{k}$ on $U$ and every multi-index $\alpha$ of length $n$ with $|\alpha|\le k$, the function $\partial^{\alpha}g$ exists on $U$ and is continuous at every point of $U$. We argue by induction on $k$.

$Q(1)$. Let $g$ be of class $C^{1}$ on $U$ and $|\alpha|\le 1$. If $\alpha=0$, then $\partial^{\alpha}g=g$ by clause 1 of Partial Derivative of a Multi-Index on a Euclidean Open Set, and $g$ is continuous at every point of $U$ by clause 1 of C^k Maps on a Euclidean Open Set. Otherwise $|\alpha|=1$, so $\alpha=e_i$ for some $i$ with $1\le i\le n$ by observation (b), and $i$ is the least index at which $\alpha$ is nonzero. Now $\partial^{\alpha-e_i}g=\partial^{0}g=g$ exists on $U$, and by clause 1 of C^k Maps on a Euclidean Open Set the partial derivative of $g$ with respect to the $i$th variable exists at every point of $U$ and the function $\partial_i g$ is continuous at every point of $U$. By clause 2 of Partial Derivative of a Multi-Index on a Euclidean Open Set, $\partial^{\alpha}g$ exists on $U$ and equals $\partial_i g$, which is continuous at every point of $U$.

$Q(k)$ implies $Q(k+1)$. Let $g$ be of class $C^{k+1}$ on $U$ and let $|\alpha|\le k+1$. If $|\alpha|\le k$, then $g$ is of class $C^{k}$ on $U$ by claim 2 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, and $Q(k)$ gives the conclusion. So assume $|\alpha|=k+1$; then $\alpha\ne 0$, and we take $i$ as in observation (a). By clause 2 of C^k Maps on a Euclidean Open Set, $g$ is of class $C^{1}$ on $U$ and $\partial_i g$ is of class $C^{k}$ on $U$; in particular, by clause 1 of that definition, the partial derivative of $g$ with respect to the $i$th variable exists at every point of $U$, so $\partial_i g:U\to\mathbb{R}$ is defined. By observation (b), $|\alpha-e_i|=k$, so $Q(k)$ applied to $\partial_i g$ and $\alpha-e_i$ gives that $\partial^{\alpha-e_i}(\partial_i g)$ exists on $U$ and is continuous at every point of $U$. The hypotheses of Peeling the Innermost Variable from a Multi-Index Partial Derivative are now met for $g$, $\alpha$ and $i$, so $\partial^{\alpha}g$ exists on $U$ and $\partial^{\alpha}g=\partial^{\alpha-e_i}(\partial_i g)$; hence $\partial^{\alpha}g$ is continuous at every point of $U$.

By induction $Q(k)$ holds for every natural number $k$, and claim 1 follows.

Claim 2. We prove by induction on the natural number $k$ the statement $R(k)$: for every $g:U\to\mathbb{R}$ smooth on $U$ and every multi-index $\alpha$ of length $n$ with $|\alpha|\le k$, the function $\partial^{\alpha}g$ exists on $U$ and is smooth on $U$.

$R(1)$. Let $g$ be smooth on $U$ and $|\alpha|\le 1$. If $\alpha=0$, then $\partial^{\alpha}g=g$ is smooth on $U$ by hypothesis. Otherwise $\alpha=e_i$ by observation (b). By claim 3 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, applied to $g$ regarded as a map into $\mathbb{R}^{1}$, the partial derivative of $g$ with respect to the $i$th variable exists at every point of $U$ and $\partial_i g$ is smooth on $U$. As in the proof of $Q(1)$, clause 2 of Partial Derivative of a Multi-Index on a Euclidean Open Set gives that $\partial^{\alpha}g$ exists on $U$ and equals $\partial_i g$, which is smooth on $U$.

$R(k)$ implies $R(k+1)$. Let $g$ be smooth on $U$ and $|\alpha|\le k+1$. If $|\alpha|\le k$, apply $R(k)$. Otherwise $|\alpha|=k+1$, so $\alpha\ne 0$; take $i$ as in observation (a). By claim 3 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, $\partial_i g$ exists on $U$ and is smooth on $U$, and by observation (b), $|\alpha-e_i|=k$. So $R(k)$ applied to $\partial_i g$ and $\alpha-e_i$ gives that $\partial^{\alpha-e_i}(\partial_i g)$ exists on $U$ and is smooth on $U$. By Peeling the Innermost Variable from a Multi-Index Partial Derivative, $\partial^{\alpha}g$ exists on $U$ and equals $\partial^{\alpha-e_i}(\partial_i g)$, hence is smooth on $U$.

Finally, let $F$ be smooth on $U$ and let $\alpha$ be an arbitrary multi-index of length $n$. By claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, every coordinate function $F_j$ is smooth on $U$. If $\alpha=0$, then $\partial^{\alpha}F_j=F_j$ is smooth on $U$ for every $j$; otherwise $|\alpha|$ is a natural number by observation (b), and $R(|\alpha|)$ applied to each $F_j$ shows that $\partial^{\alpha}F_j$ exists on $U$ and is smooth on $U$. In either case, by the final paragraph of Partial Derivative of a Multi-Index on a Euclidean Open Set, $\partial^{\alpha}F$ exists on $U$ and its coordinate functions are the smooth functions $\partial^{\alpha}F_j$; so by claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, $\partial^{\alpha}F$ is smooth on $U$.