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I know there are some sufficient conditions ensuring the interchanging of differentiation and limits in the one-dimensional casel

I am curios if one can put enough sufficient condition so that this holds for the multivariate case. In particular, I'd like to know what other conditions besides uniform convergence is needed (i.e., $f_n:R^n\rightarrow R$ converges uniformly to $f$ and $f_n$ is has well defined mixed partials) so that the mixed partials of $f_n$ converges to the mixed partials of $f$ pointwisely.

In other words, I would like $\frac{\partial f_n}{\partial x_i \partial x_j}$ to converge pointwisely to $\frac{\partial f}{\partial x_i \partial x_j}$

I would very much appreciate any references. Many thanks!

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With respect to what variables? – Qiaochu Yuan Aug 6 '12 at 0:39
just updated - thanks – scratchingmyhead Aug 6 '12 at 0:41

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