Consider a function $f(x,y)$ defined in an open disk $D$ around $(0,0)$. Assume that the partial derivatives $f_x, f_y, f_{xy}, f_{yx}$ exist in $D$ and that $f_{xy}, f_{yx}$ are continuous at $(0,0)$. Does this have any implication for continuity of $f$ at $(0,0)$? My guess is that $f$ doesn't have to be continuous at $(0,0)$ but I don't have a counterexample.
Some related results:
Having only the existence of $f_x, f_y$ doesn't imply continuity, this is a well-known example given by, say $f=0$ if $xy=0$ and $f=1$ otherwise.
Having only the existence of $f_x, f_y, f_{xy}, f_{yx}$ is not enough to guarantee continuity, as there exists a discontinuous function for which partial derivatives of all orders exist. What I'm interested here is what happens if we add continuity of $f_{xy}, f_{yx}$ at the point $(0,0)$.
We can use the mean value theorem to write
$$f(h,k)-f(0,0) = f(h,k) - f(0,k) + f(0,k)-f(0,0) = f_x(c_h, k) h + f_y(0,c_k) k$$
where $c_h$ is a point between $0$ and $h$, and similarly for $c_k$. In particular, if the functions $f_x, f_y$ are bounded near $(0,0)$ then we can get continuity. So if we want to construct an example for which continuity fails, $f_x, f_y$ should be unbounded near $(0,0)$.
