We define the function $$\Vert \cdot \Vert: \mathbb{R} \to \left[0,\frac 1 2\right]$$ as $$\Vert x \Vert = \min\big(x - \lfloor x\rfloor, 1 - (x - \lfloor x \rfloor)\big)$$ for $x \in \mathbb{R}$. I proved that the triangular inequality is true (the proof is quite simple, but a bit long because I deal with five different sub cases, I have written in italian so I do not post it, if you want I can write it). We know that $\Vert \cdot \Vert$ is not a norm nor a semi norm because it is not true that, for $\lambda, x \in \mathbb{R}$ it holds $$\Vert\lambda x\Vert = \vert\lambda\vert\cdot\Vert x\Vert.$$ Using the triangular inequality it follows that it is true the following: $$\Vert\lambda x\Vert \leq \vert\lambda\vert\cdot\Vert x\Vert.$$
I use the symbol "$\Vert$" to denote $\Vert\cdot\Vert$ because it is a norm in $\mathbb{R}/\mathbb{Z}$ but I think that this notation is a bit weird. My question is: is that function a kind of norm in $\mathbb{R}$ (I know only norms and semi norms, maybe there are other types of norm)? I would like to know this because otherwise I would have better to change the way I denote that function, using the standard notation for function, such as $f(\cdot)$ instead of $\Vert\cdot\Vert$.