In 1949 Tolstov proved that there exists $f:\mathbb{R^2} \to \mathbb{R}$ so that $f_x$ and $f_y$ exist and are continuous everywhere and $f_{xy}$ and $f_{yx}$ exist everywhere and differ on a set of positive Lebesgue measure. I have also heard that he provided an example where the cross partials do not exist everywhere but differ on a set of full measure. The only copies of the papers I have heard of are in Russian so I could not read them to find his examples. In any case I would still like to play around with those questions a bit more on my own before looking up the answers.
I am wondering if any further progress on this question is reasonably well known? It seems like this question is natural enough that it should either be a known open research problem or solved, though I may be wrong on that. My intuition says that if the cross partials can differ on a set of positive measure or even a.e. (but not necessarily exist everywhere) then it should probably be the case that they can exist and be different everywhere, though I would not be surprised to be proven completely wrong.
More broadly, if anyone has a good way of thinking about functions with cross partial derivatives which exist everywhere and are different on some "large" set I would be interested in getting a better feel for what kind of pathological behavior I should be looking for.
Best Answer
My posts in the following 2006 and 2007 sci.math threads may be of interest:
second partial derivatives commute (Clairaut's Thm.)
http://groups.google.com/group/sci.math/msg/78a769e88081ec8b
http://groups.google.com/group/sci.math/msg/3b7eeedf51fc5b92
http://groups.google.com/group/sci.math/msg/cd695899296573b6
History of mixed partial derivatives equality
http://groups.google.com/group/sci.math/msg/14aa228dadc9ed5b