$(3$ points$)$Consider the function $f(x,y)=xy*\frac{x^2-y^2}{x^2+y^2}$ which is continuous everywhere but $(0,0).$

$($a$)$ Find a value $L$ so that defining $f(0,0)=L$ makes $f(x,y)$ continuous everywhere.

$($b$)$ Find $de{l}_{x}f(x,y)$ and $de{l}_{y}f(x,y).$

$($c$)$ Find $de{l}_{x}f(0,0)$ and $de{l}_{y}f(0,0)$ using the formal limit definition of the derivative.