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

$F(x,\theta(z,x))=z$ 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 $z\in\mathbb R.$

Note that this needs a version of the Implicit Function Theorem which is "doubly global", i.e. for fixed $z\in\mathbb R$ we want the global existence of an implicit solution to the equation $F(x,y)=z$ and second we want that for every $z\in\mathbb R$. 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 $F$. References are welcome...

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

$F(x,\theta(z,x))=z$ 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 $z\in\mathbb R.$

Local results can be found for example here, which allows for quite general functions $F$. References are welcome...

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 $F$.

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

$F(x,\theta(z,x))=z$ 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 $z\in\mathbb R.$

Note that this needs a version of the Implicit Function Theorem which is "doubly global", i.e. for fixed $z\in\mathbb R$ we want the global existence of an implicit solution to the equation $F(x,y)=z$ and second we want that for every $z\in\mathbb R$. 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 $F$. References are welcome...