  This paper discusses the possibility to find and construct \textit{piecewise
constant martingales}, that is, martingales with piecewise constant sample
paths evolving in a connected subset of $\mathbb{R}$. After a brief review of
standard possible techniques, we propose a construction based on the sampling
of latent martingales $\tilde{Z}$ with \textit{lazy clocks} $\theta$. These
$\theta$ are time-change processes staying in arrears of the true time but that
can synchronize at random times to the real clock. This specific choice makes
the resulting time-changed process $Z_t=\tilde{Z}_{\theta_t}$ a martingale
(called a \textit{lazy martingale}) without any assumptions on $\tilde{Z}$, and
in most cases, the lazy clock $\theta$ is adapted to the filtration of the lazy
martingale $Z$. This would not be the case if the stochastic clock $\theta$
could be ahead of the real clock, as typically the case using standard
time-change processes. The proposed approach yields an easy way to construct
analytically tractable lazy martingales evolving on (intervals of)
$\mathbb{R}$.
