The function $q$ 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.