Consider the function defined by F(x, y) = xy(2x^2 + 6y^2)/x^2 + y^2
except at (x, y) = (0, 0) where F(0, 0) = 0. Then we have
partial differential/partial differential y partial differential F/partial differential z (0, 0) =
partial differential/partial differential z partial differential F/partial differential y (0, 0) =
Note that the answers are different. The existence and continuity of all second partials in a region around a point guarantees the equality of the two mixed second derivatives at the point. In the above case, continuity fails at (0,0).