we show by an example that this notion differs to the notion of segment-dense set introduced in [@DTL]. Further, we emphasize that the local generalized monotonicity on a self segment-dense set contains as particular cases the local monotonicity conditions considered in [@KaPiLa], [@KaPiLa1] and [@La]. Section 3 contains the main results. Our analysis starts with the one dimensional case ($X=\mathbb{R}$) in which we show that a lower semicontinuous set-valued operator that is locally generalized monotone on a dense subset of its domain must be globally generalized monotone. These results are then extended to the case of arbitrary Banach spaces using the concept of a self segment-dense subset. In Section 4 the previous results are applied to obtain the global generalized convexity of locally generalized convex functions under mild assumptions. Also an example of a locally quasiconvex continuously differentiable function which is not globally quasiconvex is given.

Preliminaries
=============

In what follows $X$ denotes a real Banach space and $X^{\ast }$ denotes the topological dual of $X.$ For $x\in X$ and $x^{\ast }\in X^{\ast }$ we denote by $\<x^{\ast },x\>$ the scalar $x^{\ast }(x)$.

We will also often use the following notations for the open, respectively closed, line segments in $X$ 