Question

f(x, y)=\left\{\begin{array}{cl} \frac{x y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0) \end{array}\right. (a) Show that f(x, y) is not continuous at (0,0). \text { (b) Show that } f_{x}(0,0) \text { and } f_{y}(0,0) \text { exist. } \text { (c) Show that } f_{x} \text { and } f_{y} \text { are not differentiable at }(0,0) \text { . }

Fig: 1

Fig: 2

Fig: 3

Fig: 4