$f(x,y)$ is defined as
$$\displaystyle f(x,y)=\begin{cases} \displaystyle x^2\sin \left(\frac{1}{x}\right) +y^2 & \mbox{if }x\neq0 \\ y^2 & \mbox{if }x=0 \end{cases}$$
Determine $f_x$ and $f_y$ at $(0,0)$ using definition of partial derivative.
Determine $f_x$ is not continuous at $(0,0)$.