I asked this question on MSE [here](https://math.stackexchange.com/questions/3437400/does-clairaut-schwarz-theorem-hold-when-mixed-partial-derivatives-are-of-order-g). One person gave an answer but then he deleted it because my version of Clairaut-Schwarz theorem is stronger than his. I meant my version only requires the continuity of **one** mixed partial derivative while his may requires the continuity of **all** mixed partial derivatives.


It seems that this question will receive no answer in MSE, so I have no choice but to post it on *mathoverflow.net*.




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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 $$\partial\_i \partial\_j f (a) = \partial\_j \partial\_i f (a)$$
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I would like to ask if Clairaut-Schwarz theorem holds in case the mixed partial derivatives are of arbitrary order $m$, i.e.



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> 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 $$\partial\_{j\_1} \partial\_{j\_2} \cdots \partial\_{j\_m} f (a)= \partial\_{j\_{\sigma(1)}} \partial\_{j\_{\sigma(2)}} \cdots \partial\_{j\_{\sigma(m)}} f(a)$$
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Thank you so much for your help!



Does this version of Clairaut-Schwarz theorem hold when mixed partial derivatives are of order greater than $2$?
Assume $f:X\to F$ has the partial derivatives $\partial\_m\partial\_{m-1}\dots\partial\_1f(x)$ at any point $x$ of the open rectangle $X:=\prod\_{1\le i\le n}]b\_i,c\_i[$.


For $u\in \mathbb{R}^n$, let $\delta\_u $ denote the finite difference $\delta\_uf(x):=f(x+u)-f(x)$, defined for $x$ and $x+u\in X$, and let $(e\_i)\_i$ be the standard basis of $\mathbb{R}^n$. 


Iterating the Mean Value Theorem, we have, say with $|t\_j|<r$ and $[x\_j-r,x\_j+r]\subset\; ]b\_j,c\_j[$ for $j=1\dots m$
$$\big\|\delta\_{t\_me\_m}\dots \delta\_{t\_1e\_1}f(x)\big\|\_F\le |t\_1\dots t\_m| \sup\_{|x\_j-y\_j| \le r\atop 1\le j\le m}\big\|\partial\_m \dots\partial\_1f(y) \big\|\_F.$$ 


If $p\in F$ and we apply the latter to the function $g(x):=f(x)-px\_1\dots x\_m $ we also get
$$\big\|\delta\_{t\_me\_m}\dots \delta\_{t\_1e\_1}f(x)-p\,t\_1\dots t\_m\big\|\_F\le |t\_1\dots t\_m| \sup\_{|x\_j-y\_j| \le r\atop 1\le j\le m}\big\|\partial\_m \dots\partial\_1f(y) -p\big\|\_F.$$  

In particular if we take $p:=\partial\_m \dots\partial\_1f(a)$, and we assume that $\partial\_m \dots\partial\_1f$ is continuous at $x=a$, we have 


$$\big\|\frac{ \delta\_{t\_me\_m}\dots \delta\_{t\_1e\_1}f(a)} {t\_1\dots t\_m} -\partial\_m \dots\partial\_1f(a)\big\|\_F=o(1) $$ for $\|t\|\to0$. Assume now there exists the $(m-1)$-order partial derivative $\partial\_{\sigma\_{m-1}}\dots\partial\_{\sigma\_1}f(x)$ for all $x\in X$ and for a given permutation $\sigma$ of $\{1,\dots,m\}$. Since all $\delta\_ {t\_j e\_j}$ commute, taking the limit for $ t\_{\sigma\_{j}}\to0$ from $j=1$ to $j=m-1$, we arrive at  

$$\big\|\frac{\partial\_{\sigma\_{m-1}}\dots\partial\_{\sigma\_1}f(a+t\_{\sigma\_m})-\partial\_{\sigma\_{m-1}}\dots\partial\_{\sigma\_1}f(a)} {t\_{\sigma\_m}} -\partial\_m \dots\partial\_1f(a)\big\|\_F=o(1) $$ for $t\_{\sigma\_m}\to0,$ 
whence the thesis: there exists $\partial\_{\sigma\_{m}}\dots\partial\_{\sigma\_1}f$ at $x=a$ and coincides with $\partial\_m \dots\partial\_1f(a)$.


(Note: To simplify the notation I assumed $(j\_1,\dots,j\_m)$ are $(1,\dots,m)$; nothing changes in general even if some of the $j\_k$ coincide, because any mixed derivative can be written $\partial\_{j\_m}\dots\partial\_{j\_1}f(a)=\frac{\partial }{\partial s\_m}\dots\frac{\partial }{\partial s\_1}f(a+s\_1e\_{j\_1}+\dots+s\_m e\_{j\_m})|\_{s\_1=0\dots s\_m=0}$). 


