Mixed derivatives in multivariable calculus Suppose $f$ is a function having variables $x$ and $y$. Can $f_{xy}$ at a point be equal to $f_{yx}$ without any of the two being continuous there? If they are equal but discontinuous functions? 

The function
$$f(x,y) = \begin{cases}
                     \frac{2x^{2}y}{x^4+y^2} & \mbox{ for } (x, y) \ne (0, 0)\\
                      0                            & \mbox{ for } (x, y) = (0, 0).
                \end{cases}$$
is a good example of a non-continuous function whose partial derivatives all exist.
