with the endpoints $x$ and $y$ $$\begin{aligned}
(x,y) &=&\big\{z\in X:z=x+t(y-x),\,t\in (0,1)\big\}, \\
\lbrack x,y] &=&\big\{z\in X:z=x+t(y-x),\,t\in \lbrack 0,1]\big\}.\end{aligned}$$

For a non-empty set $D\subseteq X$, we denote by $\operatorname*{int}(D)$ its interior and by $\operatorname*{cl}(D)$ its closure. We say that $P\subseteq D$ is dense in $D$ if $D\subseteq \operatorname*{cl}(P)$, and that $P\subseteq X$ is closed regarding $D$ if $\operatorname*{cl}(P)\cap D=P\cap D.$

Let $T:X\rightrightarrows X^{\ast }$ be a set-valued operator. We denote by $D(T)=\{x\in X:T(x)\neq \emptyset \}$ its domain and by $R(T)=\displaystyle\bigcup_{x\in D(T)}{T(x)}$ its range. The graph of the operator $T$ is the set $G(T)=\{(x,x^{\ast })\in X\times X^{\ast }:\,x^{\ast }\in T(x)\}.$

Recall that $T$ is said to be *upper semicontinuous* at $x\in D(T)$ if for every open set $N\subseteq X^{\ast }$ containing $T(x)$, there exists a neighborhood $M\subseteq X$ of $x$ such that $T(M)\subseteq N.$ $T$ is said to be *lower semicontinuous* at $x\in D(T)$ if for every open set $N\subseteq X^{\ast }$ satisfying $T(x)\cap N\neq \emptyset $, there exists a neighborhood $M\subseteq X$ of $x$ such that for every $y\in M\cap D(T)$ one has $T(y)\cap N\neq \emptyset .$ $T$ is upper semicontinuous (lower semicontinuous) on $D(T)$ if it is upper semicontinuous (lower semicontinuous) at every $x\in D(T).$

It can easily be observed 