 Solution

For each fixed $(x,y)$, the function $z\mapsto z^3+z+xy-1$ is continuous, strictly increasing because its derivative is $3z^2+1>0$, and tends to opposite infinities at the two ends of the real line. The intermediate value theorem therefore gives exactly one zero.

For $F(x,y,z)=z^3+z+xy-1$, its partial derivative $F_z=3z^2+1$ never vanishes. The implicit function theorem gives a local $C^1$ solution near every point of the solution set. Uniqueness makes these local solutions agree, producing a global $g\in C^1({\mathbb{R}}^2)$. Implicit differentiation givesAt $(1,1)$ the root is $g(1,1)=0$, so