  In this paper we study $(\epsilon,\delta)$-lc singularites, i.e.
$\epsilon$-lc singularities admitting a $\delta$-plt blow-up. We prove that
$n$-dimensional $(\epsilon,\delta)$-lc singularities are bounded up to a
deformation, and $2$-dimensional $(\epsilon,\delta)$-lc singularities form a
bounded family. Furthermore, we give an example which shows that
$(\epsilon,\delta)$-lc singularities are not bounded in higher dimensions, even
in the analytic sense.
