let $f:I\longrightarrow\mathbb{R}$ be a function.

-   [(see Lemma 2.1, [@KaPiLa1])]{} If $f$ is locally increasing on $I$, then $f$ is globally increasing on $I$.

-   [(see Theorem 2.2, [@La])]{} If $f$ is locally strictly quasimonotone on $I$, then $f$ is globally strictly quasimonotone on $I$.

-   [(see Theorem 2.3, [@La])]{} If $f$ is locally pseudomonotone on $I$, then $f$ is globally pseudomonotone on $I$.

-   [(see Theorem 2.4, [@La])]{} If $f$ is locally strictly pseudomonotone on $I$, then $f$ is globally strictly pseudomonotone on $I$.

However, local quasimonotonicity does not imply global quasimonotonicity as the next example shows.

(Example 2.6, [@La]) Let $f:{{\mathbb R}}\longrightarrow{{\mathbb R}},\,$ $$f(x)=\left\{
\begin{array}{lll}
-x-1,\,\mbox{if\,}\, x<-1\\
0,\,\mbox{if\,}\, x\in[-1,1]\\
-x+1,\,\mbox{if\,}\,x>1.\\
\end{array}
\right.$$ We can see that $f$ is continuous and locally quasimonotone on ${{\mathbb R}}$ but $f$ is not globally quasimonotone.

Next we extend the statements $(a)-(d)$ in Proposition \[p21\] to the case of set-valued operators. We need the following notion.

Let $Z$ and $Y$ be two arbitrary sets. A single valued selection of the set-valued map $F:Z\rightrightarrows Y$ is the single valued map $f:Z\longrightarrow Y$ satisfying $f(z)\in F(z)$ for all $z\in Z.$

\[p31\] Let $I\subseteq {\mathbb{R}}$ be an interval and let $F:I\rightrightarrows {\mathbb{R}}$ be a set-valued function. 