is lower semicontinuous at a point of its domain must be single-valued at that point.

Furthermore, Proposition \[ppp\] does not remain valid for set-valued upper semicontinuous operators (see [@KaPiLa], Example 2). Next we also provide an example of an upper semicontinuous set-valued operator which is strictly pseudomonotone on a dense subset of its open and convex domain, but which is not even quasimonotone on its domain.

\[ex2\] Consider the operator $$F:\mathbb{R} \rightrightarrows\mathbb{R} ,\, F(x):=\left\{
\begin{array}{ll}
x+2,\mbox{ if } x\in{\mathbb{R}}\setminus\{0\}, \\
\, [-1,3],\mbox{ if } x=0.
\end{array}
\right.$$

Let $V:=\{0\}.$ Then, $F$ is upper semicontinuous on $D(F)={\mathbb{R}}$ and strictly pseudomonotone on $D(F) \setminus V,$ but is not even quasimonotone on $D(F).$\
Obviously $D(F)\setminus V={\mathbb{R}}\setminus\{0\}$ is dense in $D(F)={\mathbb{R}}.$ Since $F(x)=x+2$ on $D(F)\setminus V,$ it is obvious that there is strictly pseudomonotone. It can easily be verified that $F$ is upper semicontinuous on ${\mathbb{R}}.$ But, for $(x,x^*)=(-1,1)\in G(F)$ and $(y,y^*)=(0,-1)\in G(F),$ we get $\langle x^*,y-x\rangle=1>0$ and $\langle y^*,y-x\rangle=-1<0,$ which shows that $F$ is not quasimonotone on ${\mathbb{R}}.$

According to Example \[ex1\], Example \[ex2\] and Remark \[r22\] we can obtain global generalized monotonicity results for a set-valued operator, based on its local appropriate generalized monotonicity property, only in the case when the operator 