Is there example of let's say a two variable function that can be rewritten as an explicit function of one variable but that does not satisfy the assumptions of the implicit function theorem?
In fact, if the theorem only provides sufficient conditions, we should say that there exists some implicit function on which, however, the theorem may fail to be applied.
The equation $F(x,y)=(x-y)^2=0$ can be solved for $y=f(x)=x$ (a smooth function) despite $\partial F/\partial y$ being zero at every point on the curve $F=0$.