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If a function has, say, partial derivatives up to order $n$, can you conclude continuity of some or all derivatives of lower order?

Especially, if a function has partial derivatives of any order is it automatically smooth?

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Please post this question at Math.Stackexchange.com –  Chandrasekhar Oct 7 '11 at 17:01
As you wish, but it seemed right here. It seems being quite a typical question you ask when you learn about partial differentiability, but interestingly hard to answer. I couldn'd find any source that answered it (books, wikipedia), and I couldn't come up with any counterexample. –  Kofi Oct 7 '11 at 17:59
This is not a MO question. Anyway, maybe I am misunderstanding the question, but one can construct very easily a function (in two variables) which is not continuous in $(0,0)$, but it is derivable along any direction. So the answer is negative. –  Valerio Capraro Oct 7 '11 at 18:26
It is continuous if you require commutation of partial derivatives I think. –  user16974 Oct 7 '11 at 18:39
I posted the question at stackexchange, but got no response that I didn't already know. math.stackexchange.com/questions/70656/… The thing is, that it has to be differentiable in MIXED directions. –  Kofi Oct 11 '11 at 16:26
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closed as too localized by Andrey Rekalo, Deane Yang, Igor Rivin, quid, Bill Johnson Oct 7 '11 at 19:19

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