Clairaut's Theorem: Continuous 1st Derivatives?

[alexio]

If second derivatives of function,f, fxy and fyx, are equal,
are the first dervitives, fx and fy, continuous?

 

[Hootenanny]

If second derivatives of function,f, fxy and fyx, are equal,
are the first dervitives, fx and fy, continuous?

Yes. For the second order partial derivatives of a function to exist on some set, the first order partial derivatives must be differentiable and hence continuous on that same set. More interestingly, if both second order partial derivatives exist are are equal on some set, then they are both continuous on that same set. The opposite statement is also true.