Chapter 14.4, Problem 54E

(a) The function

$f(x,y)=\left\{\begin{array}{ll}\frac{xy}{x^2+y^2}\hfill & \text{ if }(x,y)\ne (0,0)\hfill \\ 0\hfill & \text{ if }(x,y)=(0,0)\hfill \end{array}\right.$

was graphed in Figure 4. Show that $f_x(0,0)$ and $f_y(0,0)$ both exist but $f$ is not differentiable at $(0,0)$ . [Hint: Use the result of Exercise 53.]

(b) Explain why $f_x$ and $f_y$ are not continuous at $(0,0)$ .