Is there a criterion for showing that a two variable continuous function $f:\Bbb S^1\times\Bbb S^1\to \Bbb R^2$ pass from the origin? Is there a criterion for showing that  a symmetric ($f(x,y)=f(y,x)$) two variable continuous function $f:\Bbb S^1\times\Bbb S^1\to \Bbb R^2$ pass from the origin? i.e. 
$$\exists\,(x,y)\in \Bbb S^1\times\Bbb S^1,\,x\neq y\quad s.t. \quad f(x,y)=(0,0).$$
  where we know that $f$ pass from all four regions and $f(x,x)=(0,0)$ for all $x\in\Bbb S^1$. I tried to define a real value function from $f$ and use from intermediate value theorem but I couldn't find a helpful map. Is it sufficient to show ${\rm Im} f$ is simply connected?
 A: Let $\alpha(\theta)=\left\lgroup
\begin{array}{c}
\cos \theta
\\
\sin \theta
\end{array}
\right\rgroup \in \mathbb{S}^1$ and $\beta(\varphi)=\left\lgroup
\begin{array}{c}
\cos \varphi
\\
\sin \varphi
\end{array}
\right\rgroup\in\mathbb{S}^1$ parametrizations of $\mathbb{S}^1$ whit $0\leq \theta\leq 2\cdot \pi$ and $0\leq \varphi\leq 2\cdot \pi$. 
By Poincaré-Miranda theorem if there are $\theta^\prime,\theta^{\prime\prime}\in [0,2\pi]$ and $\varphi^\prime,\varphi^{\prime\prime}\in [0,2\pi]$ such that
$$
f_1
\left(
\left\lgroup
\begin{array}{c}
\cos \theta^\prime
\\
\sin \theta^{\prime}
\end{array}
\right\rgroup
,
\left\lgroup
\begin{array}{c}
\cos \varphi
\\
\sin \varphi
\end{array}
\right\rgroup
\right)
<0,
\quad
f_1
\left(
\left\lgroup
\begin{array}{c}
\cos \theta^{\prime\prime}
\\
\sin \theta^{\prime\prime}
\end{array}
\right\rgroup
,
\left\lgroup
\begin{array}{c}
\cos \varphi
\\
\sin \varphi
\end{array}
\right\rgroup
\right)
>0,\quad \forall \varphi \in [\varphi^{\prime},\varphi^{\prime\prime}]
$$
$$
f_2
\left(
\left\lgroup
\begin{array}{c}
\cos \theta
\\
\sin \theta
\end{array}
\right\rgroup
,
\left\lgroup
\begin{array}{c}
\cos \varphi^{\prime}
\\
\sin \varphi^{\prime}
\end{array}
\right\rgroup
\right)
<0,
\quad
f_2
\left(
\left\lgroup
\begin{array}{c}
\cos \theta
\\
\sin \theta
\end{array}
\right\rgroup
,
\left\lgroup
\begin{array}{c}
\cos \varphi^{\prime\prime}
\\
\sin \varphi^{\prime\prime}
\end{array}
\right\rgroup
\right)
>0,\quad \forall \theta \in [\theta^{\prime},\theta^{\prime\prime}]
$$
then there are $\theta^\ast$ and $\varphi^\ast$ such that
$$
f_1
\left(
\left\lgroup
\begin{array}{c}
\cos \theta^{\ast}
\\
\sin \theta^{\ast}
\end{array}
\right\rgroup
,
\left\lgroup
\begin{array}{c}
\cos \varphi^{\ast}
\\
\sin \varphi^{\ast}
\end{array}
\right\rgroup
\right)
=0\;\;
\mbox{ and }
\;\;
f_2
\left(
\left\lgroup
\begin{array}{c}
\cos \theta^{\ast}
\\
\sin \theta^{\ast}
\end{array}
\right\rgroup
,
\left\lgroup
\begin{array}{c}
\cos \varphi^{\ast}
\\
\sin \varphi^{\ast}
\end{array}
\right\rgroup
\right)
=0,
$$
The Poincaré–Miranda theorem is a generalization of the intermediate value theorem.
