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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