-u^{\varepsilon }\left( y\right)
\right\vert <\varepsilon /2.$$If we take a regular grid (partition) with step $h$ small enough ($h<\delta $) and choose $u^{h}\in V^{0}\left( \left[ 0,1\right] ,\mathbb{R}\right) $ such that $u^{h}\left( \left( i-\frac{1}{2}\right) h\right) =u^{\varepsilon
}\left( \left( i-\frac{1}{2}\right) h\right) $ for $i:=0,1,\ldots \frac{1}{h}
$ then$$\left\Vert u^{h}-u^{\varepsilon }\right\Vert _{\infty }<\varepsilon /2.$$Hence$$\begin{aligned}
\left\Vert u-u^{h}\right\Vert _{2} &\leq &\left\Vert u-u^{\varepsilon
}\right\Vert _{2}+\left\Vert u^{\varepsilon }-u^{h}\right\Vert _{2} \\
&\leq &\varepsilon /2+\varepsilon /2.\end{aligned}$$

In the next section we show that a lower semicontinuous set-valued operator that possesses a local generalized monotonicity property on a self segment-dense subset of its domain is actually globally generalized monotone. According to the previous remark, this result not only extends but also improves the results presented in Proposition \[pp\].

On the local generalized monotonicity of set-valued maps on dense subsets
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In this section we show that the local strict quasimonotonicity, local pseudomonotonicity, respectively local strict pseudomonotonicity property of a lower semicontinuous set-valued operator on a self segment-dense subset of its domain provides the global appropriate monotonicity of that operator under some mild conditions. We attain this goal gradually, by showing first that the previous statement holds for real set-valued lower semicontinuous functions of one real variable. We also show that our conditions 