Is there a function $f:O\to \mathbb{R}$, $O$ is an open subset in $\mathbb{R}^2$, which satisfies both $(1)$ and $(2)$ ?

$(1)$ All of its second partial derivatives are defined on $O$ and continuous at $(x_0,y_0)\in O$ ;

$(2)$ It is not differentiable in any neighborhood of $(x_0,y_0).$

Obviously, $(1)$ involves the differentiability of $f$ at $(x_0,y_0)$.

The original post come from here. In order to express my purpose clearly, I simplify $n$-variables to two-variables.