Let $u,v\in U_x\cap I,\, u\neq v.$ Then, by the denseness of $J$ in $I$, we obtain that there exists $w\in(u,v)\cap J,$ and according to the hypothesis of the theorem $w$ admits an open neighborhood $U_w$ such that $F|_{U_w\cap J}$ is strictly quasimonotone. This in particular shows, that there exists $z\in(u,v)$ and $z^*\in F(z)$, such that $\<z^*,v-u\>\neq 0.$ Hence, $F$ is strictly quasimonotone on $U_x\cap I.$

If $F(x)=\{0\}$, then $x\in J$, hence there exists an open neighborhood $U_x$ of $x$ such that $F|_{U_x\cap J}$ is strictly quasimonotone. Note that according to Proposition \[p32\], in case that there exists an open neighborhood $U$ of $x$ such that one of the following conditions is fulfilled

-   $F(y)\ge0$ for every $y\in U\cap I,$ or,

-   $F(y)\le 0$ for every $y\in U\cap I,$ or,

-   $F(z)\le 0$ and $F(y)\ge 0$ or every $z\le x\le y,\,y,z\in U\cap
    I,$

then $F|_{U\cap I}$ is strictly quasimonotone.

Assume now, that none of the conditions $(a)-(c)$ is fulfilled. Then, for every neighborhood $U$ of $x$, there exists $y,z\in U\cap I,\, z>x,\,y>x$ (or $z<x,\,y<x,$) such that $F(y)\ge 0,\, F(y)\neq\{0\}$ and $F(z)\le 0,\,F(z)\neq\{0\}.$ Let $y,z\in U_x\cap I,\, z>x,\,y>x,$ (the case $y,z\in U\cap I,\, z<x,\,y<x$ can be treated similarly). Obviously, we can assume 