I have an idea on this one, tell me what you guys think.
(1) Find conditions on the function f which will allow you to solve the equation f(f(x,y),y) = 0 for y as a function of x near (0,0).
I think all that is required of f is that:
(1) f is continuous and differentiable
(2) the partial of f with respect to y is non-zero, which will make the required determinant non-zero for use in the implicit function theorem. This just seems too easy. Any thoughts?