When does Order of Second Partial Derivatives Matter?

Question

My professor was saying that, for a function of multiple variables, usually the order in which you take the order of partial derivatives did not matter. (ex: $f_{xy} = f_{yx}$).

Under what circumstances is this not true?

Answer 1

It may not be true if the function is not of class $C^2$. You can check that $$f(x,y) = \begin{cases} \dfrac{xy(x^2-y^2)}{x^2+y^2}, \mbox{ if }(x,y) \neq (0,0) \\[1em] 0, \mbox{ if }(x,y) = (0,0) \end{cases}$$is a counter-example. See more details in the relevant Wikipedia page, for example.