Let $F:\mathbb R^2\rightarrow\mathbb R$ be a measurable function. Under what conditions on $F$ does there exist a function $\theta:\mathbb R^2\rightarrow\mathbb R$ satisfying

$F(x,\theta(z,x))=z$ for all $(x,y,z)\in\mathbb R^3$ with $F(x,y)=z$,

$\theta(\cdot,x):\mathbb R\rightarrow\mathbb R$ is continuous for all $x\in\mathbb R$,

$\theta(z,\cdot):\mathbb R\rightarrow\mathbb R$ is measurable for all $z\in\mathbb R.$

I have seen some papers on Global Existence of Implicit Functions, but either I don't understand them or my question is really more general than what they prove. Local results can be found for example here, which allows for quite general functions $F$.