We define a mapping which with each function $u\in L^{\mathrm{\infty }}\left(0,T\right)$ and an admissible value of $r>0$ associates the function $\xi$ with a prescribed initial condition ${\xi }^0$ which minimizes the total variation in the $r$-neighborhood of $u$ in each subinterval $\left[0,t\right]$ of $\left[0,T\right]$. We show that this mapping is non-expansive with respect to $u$, $r$ and ${\xi }^0$, and coincides with the so-called play operator if $u$ is a regulated function.