  In this paper we prove uniform oscillation estimates on $L^p$, with
$p\in(1,\infty)$, for truncated singular integrals of the Radon type associated
with Calder\'on-Zygmund kernel, both in continuous and discrete settings. In
the discrete case we use the Ionescu-Wainger multiplier theorem and the
Rademacher-Menshov inequality to handle the number-theoretic nature of the
discrete singular integral. The result we obtained in the continuous setting
can be seen as a generalisation of the results of Campbell, Jones, Reinhold and
Wierdl for the continuous singular integrals of the Calder\'on-Zygmund type.
