KöMaL 2017 C1397

Prove that the function $f\colon \mathbb{R}\to \mathbb{R}, x\mapsto \begin{cases} 2x+2-6k, & \text{if \ } x\in \left[6k; 6k+2\right[ ,\\ \frac{x-2+6k}{2}, & \text{if \ } x\in \left[6k+2; 6k+6\right[ \end{cases} (k\in \mathbb{Z})$ (where $k$ takes every integer value) is one-to-one and equal to its own inverse.