Continuity versus partials off axes

Let $f:\mathbb{R}\setminus\{0\}\to\mathbb{R}$ and $g:\mathbb{R}\setminus\{0\}\to\mathbb{R}$ be two bounded and differentiable functions. Define $F:\mathbb{R}^{2}\to\mathbb{R}$ by $$F(x,y)=\begin{cases}xf(y)+y^{2}g(x),&xy\neq 0,\\0,&xy=0.\end{cases}$$ Consider the following statements. $P$: $F$ is continuous at $(0,0)$. $Q$: Both the partial derivatives $\dfrac{\partial F}{\partial x}$ and $\dfrac{\partial F}{\partial y}$ exist at $(1,0)$. Then