A question about mixed partial derivative

vacuum

Let there be a function f[x,y]: RxR->R

Is there any connection between the differentiability
(I am not sure that this is the right English term - I meant f[x,y]= a*dx+b*dy +something of smaller order)
and the equality fxy=fyx, where fxy means the derivative of f[x,y] first by y, and than by x ?

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Originally posted by vacuum
Let there be a function f[x,y]: RxR->R

Is there any connection between the differentiability
(I am not sure that this is the right English term - I meant f[x,y]= a*dx+b*dy +something of smaller order)
and the equality fxy=fyx, where fxy means the derivative of f[x,y] first by y, and than by x ?
The idicated cross partials have to exist of course. Usually you wouls ensure that by requiring that f be twice differentiable in both variables.

vacuum

Does that mean that double differentiability implies the fxy=fyx equality?(Or reverse plus continuity of other partial derivatives?)

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Originally posted by vacuum