I usually encounter Clairaut-Schwarz theorem where the mixed partial derivatives are of order $2$, i.e.
$\textbf{Clairaut-Schwarz Theorem:}$ Let $X$ be open in $\mathbb R^n$, $f:X \to F$, and $i, j \in\{1,\ldots,n\}$. Suppose that $\partial_j \partial_i f$ is continuous at $a$ and that $\partial_j f$ exists in a neighborhood of $a$. Then $\partial_i \partial_j f (a)$ exists and $$\partial_i \partial_j f (a) = \partial_j \partial_i f (a)$$
I would like to ask if Clairaut-Schwarz theorem holds in case the mixed partial derivatives are of arbitrary order $m$, i.e.
Let $X$ be open in $\mathbb R^n$, $f:X \to F$, and $m \in \mathbb N$. Suppose $j_1, j_2, \ldots, j_m \in\{1,\ldots,n\}$ and $\sigma$ is a permutation of $\{1, \ldots, m\}$. If $\partial_{j_1} \partial_{j_2} \cdots \partial_{j_m} f$ is continuous at $a$ and $\partial_{j_{\sigma(2)}} \cdots \partial_{j_{\sigma(m)}} f$ exists in a neighborhood of $a$, then $$\partial_{j_1} \partial_{j_2} \cdots \partial_{j_m} f (a)= \partial_{j_{\sigma(1)}} \partial_{j_{\sigma(2)}} \cdots \partial_{j_{\sigma(m)}} f(a)$$
Thank you so much for your help!