Chapter 12.4, Problem 57E

Nondifferentiability? Consider the following functions f.

a.    Is f continuous at (0, 0)?

b.    Is f differentiable at (0, 0)?

c.    If possible, evaluate f_x(0, 0) and f_y(0, 0).

d.    Determine whether f_x and f_y are continuous at (0, 0).

e.    Explain why Theorems 12.5 and 12.6 are consistent with the results in parts (a)–(d).

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