Discussion
True or false: the mixed partial derivatives of every function of two variables are equal.
Answer: A
Clairaut's theorem gives equality only when the mixed derivatives are continuous at the point. Counterexamples exist in which \( f_{xy} \) and \( f_{yx} \) differ at the origin, both being discontinuous there. Mere existence of the first derivatives, or a domain covering the plane, is not enough
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