Consider a continuous function $f : [0,1]^2 \to \mathbb R$ with $f(0,y) = 1$ and $f(1,y) = -1$. I want to show that there exists some continuous curve $$(\gamma_1, \gamma_2) = \gamma : [0,1] \to [0,1]^2$$ with $\gamma_1(0)= 0$ and $\gamma_1(1) = 1$ (so that it goes from bottom to the top). And that $f\circ \gamma$ is identically zero.
The image I have in mind is: 
I think it would be some intermediate-value thm type argument, but I don't know enough about multivariate analysis/general topology to make the argument.
