self segment-dense in $L^{2}\left( \left[ 0,1\right] ,\mathbb{R}\right) $, and for any $u,v\in V^{0}\left( \left[ 0,1\right] ,\mathbb{Q}\right) $ the segment $\left[ u,v\right] $ is not contained in $V^{0}\left( \left[ 0,1\right] ,\mathbb{Q}\right) $. However, $V^{0}\left( \left[ 0,1\right] ,\mathbb{Q}\right) $ is not segment-dense (in the sense of The Luc) in $L^{2}\left( \left[ 0,1\right] ,\mathbb{R}\right) $, as can be seen by taking $u\left( x\right) =\sqrt{2}$ for all $x\in \left[ 0,1\right] $ and for any $v\in V^{0}\left( \left[ 0,1\right] ,\mathbb{Q}\right) $ we have that $\left[
u,v\right] \cap V^{0}\left( \left[ 0,1\right] ,\mathbb{Q}\right) =\{v\}$.

The proof of the fact that $V^{0}\left( \left[ 0,1\right] ,\mathbb{R}\right)
$ is dense in $L^{2}\left( \left[ 0,1\right] ,\mathbb{R}\right) $ relies on the following classical arguments.

Let $\varepsilon >0$ be given. We have that $C_{c}^{\infty }\left( \left[ 0,1\right] ,\mathbb{R}\right) $ is dense in $L^{2}\left( \left[ 0,1\right] ,\mathbb{R}\right) $ (see for example [@B]), so for a given $u\in
L^{2}\left( \left[ 0,1\right] ,\mathbb{R}\right) $ the exists $u^{\varepsilon }\in C_{c}^{\infty }\left( \left[ 0,1\right] ,\mathbb{R}\right) $ such that$$\left\Vert u-u^{\varepsilon }\right\Vert _{2}<\varepsilon /2.$$Now since $u^{\varepsilon }$ is continuous on the compact set $[0,1]$, uniform continuity guarantees that there exists $\delta >0$ such that for any $x,y\in \left[ 0,1\right] $ with $\left\vert x-y\right\vert <\delta $ we have$$\left\vert u^{\varepsilon }\left( x\right) 