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A function $f$ that has continuous third order partial derivatives in $\mathbb{R}^n$. I'm just wondering that since the partial derivatives are continuous then the Hessian matrix is symmetric. Is that correct?

Thanks.

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    Yes. If the third order partials are continuous, then the second order partials are and so Clairaut's theorem applies -- mixed partials are equal and thus the Hessian is symmetric.2012-03-19
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    Yes, that is correct. You only need second order partials to be continuous. See http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives2012-03-19

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