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).

>>9
Sorry. You're right. As you said, the cone still isn't a valid solution though.

>>12
I think I'll give you that... yeah.

The example I was thinking of was
f(x,y) = x^(3/2)sin(1/x) + y^(3/2)sin(1/y)
f(0,y) = y^(3/2)sin(1/y)
f(x,0) = x^(3/2)sin(1/x)
f(0,0) = 0

If you take f(x) = x^(3/2)sin(1/x) then, by taking the required limit, you find the derivative at 0 to be 0. However, the derivative is 3/2x^(1/2)sin(1/x) - x^(-1/2)cos(1/x) which gives an unbounded value as you approach zero.