Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice differentiable?



Function with all but mixed second partial derivatives twice differentiable?
Here is an explicit counterexample in $B\_1$,
$$f(x,y) = x\, y \, \log(-\log(x^2+y^2))$$


Both $\partial\_{xx} f$ and $\partial\_{yy} f$ are continuous but $f$ is not twice differentiable at the origin.


