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

Let $n$ and $m$ be natural numbers, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$, let $F=(F_1,\dots,F_m):U\to\mathbb{R}^{m}$ have coordinate functions $F_j:U\to\mathbb{R}$, and let $\alpha$ be a multi-index of length $n$, of order $|\alpha|$. Index ranges such as $1\le j\le m$, and the comparison of $|\alpha|$ with a natural number, use the order on the natural numbers. Multi-index partial derivatives are those of Partial Derivative of a Multi-Index on a Euclidean Open Set. Then the following hold.

1. ($C^{k}$ maps) Let $k$ be a natural number, and assume that $F$ is of class $C^{k}$ on $U$ and that $|\alpha|\le k$. Then $\partial^{\alpha}F$ exists on $U$ and, for every $j$ with $1\le j\le m$, the function $\partial^{\alpha}F_j:U\to\mathbb{R}$ is continuous at every point of $U$.

2. (Smooth maps) Assume that $F$ is smooth on $U$. Then $\partial^{\alpha}F$ exists on $U$ and the map $\partial^{\alpha}F:U\to\mathbb{R}^{m}$ is smooth on $U$.