Let $f : \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x,y) = \begin{cases} \frac{x^2 y^2}{x^4 + y^4}, & \text{ }\text{(x,y)} \neq (0,0) \\ 0, & \text{ }\text{(x,y)} = (0,0) \end{cases} .$
Show that $\frac{df}{dx} (0,0)$ exists, and $f$ is not continuous at $(0,0)$.