Example of quasi-monotone map that is not pseudo-monotone?

Question

A quasi-monotone map is a function $F$ such that for all $x,y$, \begin{equation} F(x) \cdot (y-x) > 0 \implies F(y) \cdot (y-x) \geq 0 \end{equation}

A pseudo-monotone map is a function $F$ such that for all $x,y$, \begin{equation} F(x) \cdot (y-x) \geq 0 \implies F(y) \cdot (y-x) \geq 0 \end{equation}

It is not clear to me why the quasi-monotone maps contain the set of pseudo-monotone maps.

Can someone provide an example showing that you can have a quasi-monotone map that is not pseudo-monotone?

Answer 1

In simpler terms, quasi-monotone means "if $F(x) > 0$ then $F(y) \geq 0$ for $y > x$, and if $F(x) < 0$ then $F(y) \leq 0$ for $y < x$".

Pseudo-monotone is the same, but with $F(x) \geq 0$ and $F(x) \leq 0$ in conditions respectively.

So, quasi-monotone doesn't care what happens left or right of the points where $F$ is zero, while pseudo-monotone does.

This immediately gives as an example we need:

$$F(x) = \begin{cases} -1,& x \in (-\infty, 0) \cup (0, 1)\\ 1,& x \geq 1\\ 0,& x = 0 \end{cases}$$

We then have $F(0) \cdot (\frac{1}{2} - 0) = 0 \geq 0$ but $F(\frac{1}{2}) \cdot (\frac{1}{2} - 0) = -\frac{1}{2} \not \geq 0$.