I am not sure if the use of mean value theorem for functionals is appropriate in this case but I have an intuition that it might work, although I am not sure.
The problem is as follows: Consider an implicit function $F(x,y,f)=0$, where $x,y\in \mathbb{R}$, $f:\mathbb{R}\longrightarrow \mathbb{R}$ and $f$ is continuous. I have that $\forall f\in \mathbb{R}^\mathbb{R}$, $x=y$ is never a solution to $F$. Also, there exists at least some $f$ for which $x>y$ is a solution to $F$. Can I use some version of the mean value theorem for functionals to show that $x<y$ can never be a solution to $F$, $\forall f\in \mathbb{R}^\mathbb{R}$?