In this part, we introduce the function $q$ which associates to any real $x$ the real number $q(x) = x - \lfloor x \rfloor - \frac{1}{2}$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that $q$ is piecewise continuous on $\mathbf{R}$, that it is 1-periodic and that the function $|q|$ is even.