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Formally, let $$D=[-1,1;-1,1]\subset\mathbb{R}^2,$$ and let $f,g:[0,1]\to D$ be two continuous functions, such that $f(0)=(-1,0)$, $f(1)=(1,0)$, $g(0)=(0,-1)$, $g(1)=(0,1)$. Prove that $\exists\zeta,\xi$, such that $f(\zeta)=g(\xi)$.

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see mathoverflow.net/questions/35514 –  Martin Mar 3 '13 at 0:38
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