Monsieur Ejemplé

Give an example of a function f:R^2 --> R where both partial derivatives df/dx and df/dy exist non-trivially, but neither are continuous at (0,0).

>>5

Not quite. You've got a basic idea that will work with some refinement, though. Remember (I should have made this clearer) the function is continuous at (0,0) and both partial derivatives exist at (0,0).