Since the partial derivatives are continuous, the function is differentiable. Further, the limits, as (x, y) go to 0, are 0.
So the only question is at (0,0).
Since those match the limits of the partial derivatives as (x, y) goes to (0,0), the function is also differentiable at (0,0).
Again, that shows that the partial derivatives exist and are continuous at (0,0). That implies that the function is differentiable at (0,0).