If $f $ is differentiable at $(x,y)$ then $ f_{xy}$ exists at $(x,y)$?

Question

Suppose $f:{\bf R}^2 \rightarrow {\bf R}$ is once differentiable at a point $p$. Does it follow that $f_{xy}$ (the derivative of $f $ w.r.t to $x$ and then w.r.t to $y$) exist at $p$?

Answer 1

No, try $f(x,y)=x\cdot|y|$ at $p=(0,0)$.