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If $f_i$ are real analytic functions on $\mathbb R^n$ such that for arbitrary partial derivative index $\alpha \ge 0$, $f_i^\alpha \to {f^\alpha }$ uniformly, is it necessary that $f$ is an analytic function?

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You say that $\alpha$ is just an index (and I guess, not a power) - then I wail to find how is is used in your statement. –  Ilya Dec 6 '11 at 11:41
Are you using $f^{\alpha}$ as notation for a partial derivative of $f$ of arbitrary order? If not, I, too, am puzzled as to how to understand your notation. –  Gerry Myerson Dec 6 '11 at 11:50
Sorry for the unclarity, I have clarified it. –  Hezudao Dec 6 '11 at 12:04
In mathoverflow this question was discussed. mathoverflow.net/questions/53557/… –  bonnnnn2010 Dec 6 '11 at 13:29

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