Consider the continuous and injective mapping
\begin{eqnarray*} \varphi:[0,1] &\rightarrow& \mathbb{R}^2, \\ t &\mapsto& (x(t),y(t)), \end{eqnarray*} such that $x(0)<x(1)$, and \begin{equation*} \big( (x(t)-x(s)\big)\big( y(t)-y(s)\big) \ge 0,\quad \forall t,s\in [0,1]. \end{equation*}

My intuition is that $x(0)\le x(t)\le x(1)$ for any $0<t<1$.

I have tried to use the Intermediate Value Theorem to get the result, but still cannot proceed with it.

Thank you for your reading. Any help is very appreciated.

Consider the continuous and injective mapping
\begin{eqnarray*} \varphi:[0,1] &\rightarrow& \mathbb{R}^2, \\ t &\mapsto& (x(t),y(t)), \end{eqnarray*} such that $x(0)<x(1)$, and \begin{equation*} \big( (x(t)-x(s)\big)\big( y(t)-y(s)\big) \ge 0,\quad \forall t,s\in [0,1]. \end{equation*}

My intuition is that $x(0)\le x(t)\le x(1)$ for any $0<t<1$.

I believe the key idea to solve this is to use the Intermediate Value Theorem, and the following result for univariate functions (continuity and injectivity together deduce monotonicity in, see https://math.stackexchange.com/questions/170147/a-continuous-injective-function-f-mathbbr-to-mathbbr-is-either-strict) to get the result, but still cannot proceed with it.

Thank you for your reading. Any help is very appreciated.