# Math Help - Intermediate Value Theorem for Two Dimensional Case

1. ## Intermediate Value Theorem for Two Dimensional Case

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 $f\in C([0,1]^{2},\mathbb{R})$ be increasing in both of its variables and that $f(0,0)<0$ and $f(1,1)>0$.
Then, there exists $\alpha\in C([0,1],[0,1]^{2})$ such that $f\circ\alpha=0$ on $[0,1]$.

Thanks.