Dear all, I wonder to know whether any of you have seen a result similar to that I write below.

If so please redirect me to its proof, I just wonder to see.

Theorem. Let $\displaystyle f\in C([0,1]^{2},\mathbb{R})$ be increasing in both of its variables and that $\displaystyle f(0,0)<0$ and $\displaystyle f(1,1)>0$.

Then, there exists $\displaystyle \alpha\in C([0,1],[0,1]^{2})$ such that $\displaystyle f\circ\alpha=0$ on $\displaystyle [0,1]$.

Thanks.