I have a function $F(x,y)=z$ and two points $(x_1,y_1),(x_2,y_2)$ s.t. $F(x_1,y_1)=F(x_2,y_2)=c$, $x_1<x_2$. I know that $\frac{\partial F}{\partial y}<0$ in $ [x_1,x_2]\times\mathbb{R}$.
I'd like to prove that there is a continuous contour line between the two points.
I know that there's a rectangle $V\times W $ that contains $(x_1,y_1)$ s.t. $F^{-1}(c)\cap V\times W $ is the graphic of a function, i.e., I have in there a continuous contour line.
I'd like to know if the fact that I have $\frac{\partial F}{\partial y}<0$ in the entire interval $[x_1,x_2]$ allows that I consider $V=[x_1,x_2]$ and $W=\mathbb{R}$, so I can prove the statement.
Many thanks!
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Edit for comment on 12/26