I've asked this question on MSE here. A person gave an answer but then deleted it because my version of Clairaut-Schwarz's theorem is stronger than his. I meant that my version only requires the continuity of **a** mixed partial derivative while its may require continuity of **all** partial mixed derivatives.

It seems that this question will not receive any answer in MSE, so I can only post on *mathoverflow.net*.

I usually meet the theorem of Clairaut-Schwarz where the mixed partial derivatives are of order $ 2 $, that is to say.

$ textbf {Clairaut-Schwarz theorem:} $Let $ X $ to be open in $ mathbb R ^ n $, $ f: X to F $, and $ i, j in {1, ldots, n } $. Assume that $ partial_j partial_i f $ is continuous to $ 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 the Clairaut-Schwarz theorem is valid in the case where the mixed partial derivatives are of arbitrary order $ m $, that is to say.

Let $ X $ to be open in $ mathbb R ^ n $, $ f: X to F $, and $ m in mathbb N $. assume $ j_1, j_2, ldots, j_m in {1, ldots, n } $ and $ sigma $ is a permutation of $ {1, ldots, m } $. Yes $ partial_ {j_1} partial_ {j_2} cdots partial_ {j_m} f $ is continuous to $ 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 very much for your help!