Chapter 14.4, Problem 46E

(a) The function

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

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 45.]

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