our investigations in Section 3.

\[p\]*(Theorem 3.4, [@KaPiLa1])* Let $X$ be a real Banach space and let $D\subseteq X$ be open and convex. Let $C\subseteq D$ be a set closed regarding $D,$ with empty interior, satisfying the following condition $$\forall\,x,y\in D\setminus C \mbox{ the set } [x,y]\cap C \mbox{ is countable, possibly empty.}$$ Assume that $T:D\longrightarrow X^*$ is a single-valued operator. If $T$ is continuous and the restriction $T|_{D\setminus C}$ is locally monotone, then $T$ is monotone on $D$

\[pp\] Let $X$ be a real Banach space and let $D\subseteq X$ be open and convex. Let $C\subseteq D$ be a set closed regarding $D,$ with empty interior, satisfying the following condition $$\forall\,x,y\in D\setminus C \mbox{ the set } [x,y]\cap C \mbox{ is countable, possibly empty.}$$ Assume that $T:D{\longrightarrow}X^*$ is a hemicontinuous operator with the property that $\<T(z),y-x\>\neq 0$ for all $z\in [x,y]\cap C,\,x,y\in D,\, x\neq y.$ Then the following hold.

-   *(Theorem 3.6, [@La])* If $T|_{D\setminus C}$ is locally strictly quasimonotone, then $T$ is strictly quasimonotone on $D.$

-   *(Theorem 3.7, [@La])* If $T|_{D\setminus C}$ is locally pseudomonotone, then $T$ is pseudomonotone on $D.$

-   *(Theorem 3.8, [@La])* If $T|_{D\setminus C}$ is locally strictly pseudomonotone, then $T$ is strictly pseudomonotone 