is lower semicontinuous and the mentioned monotonicity property is one of the following: strict quasimonotonicity, pseudomonotonicity or strict pseudomonotonicity, respectively.

Self segment-dense sets
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In [@DTL], Definition 3.4, The Luc has introduced the notion of a so-called *segment-dense* set. Let $V\subseteq X$ be a convex set. One says that the set $U\subseteq V$ is segment-dense in $V$ if for each $x\in V$ there can be found $y\in U$ such that $x$ is a cluster point of the set $[x,y]\cap U.$

In what follows we introduce a denseness notion, slightly different from the concept of The Luc presented above, but which is sufficient for the conditions in Proposition \[p\], Proposition \[pp\], respectively Proposition \[ppp\].

\[dd\] Consider the sets $U\subseteq V\subseteq X$ and assume that $V$ is convex.

We say that $U$ is self segment-dense in $V$ if $U$ is dense in $V$ and $$\forall x,y\in U,\mbox{  the set }[x,y]\cap U\mbox{  is dense in }[x,y].$$

Obviously in one dimension the concepts of a segment-dense set respectively a self segment-dense set are equivalent to the concept of a dense set.

Assume that $C, D,$ respectively $V$ are sets as in Proposition \[p\], Proposition \[pp\], respectively Proposition \[ppp\]. Then the set $D\setminus C$, respectively 