Can the partial derivative of f(x,y) at (a,b) exist if f(x,y) is not continuous at (a,b)? Suppose f(x,y) is continuous for all $(x,y) \neq (a,b)$, (not continuous at (a,b)), can the partial derivative with respect to x (or y) at (a,b) still exist?

Another example is 
$$f(x,y) = \begin{cases} \frac{xy}{x^2+y^2} & \text{ if}\  (x,y)\neq (0,0) \\ 0 & \text{ if}\  (x,y)=(0,0) \end{cases}$$
