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?
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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