Do there exist two functions $u,v$ defined on an open set containing $(0,0)$ with values in $\mathbb{R}$ such that
(1) $u,v$ are differentiable at $(0,0)$;
(2) $u_x=v_y, u_y=-v_x$ at $(0,0)$;
(3) at least one of $u_x,u_y,v_x,v_y$ is not continuous at $(0,0)$ ?