# Intermediate Value Theorem for Two Dimensional Case

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 $[0,1]$.