  Finding effective theories of modified gravity that can resolve cosmological
singularities and avoid other physical pathologies such as ghost and gradient
instabilities has turned out to be a rather difficult task. The concept of
limiting curvature, where one bounds a finite number of curvature-invariant
functions thanks to constraint equations, is a promising avenue in that
direction, but its implementation has only led to mixed results. Cuscuton
gravity, which can be defined as a special subclass of $k$-essence theory for
instance, is a minimal modification of gravity since it does not introduce any
new degrees of freedom on a cosmological background. Importantly, it naturally
incorporates the idea of limiting curvature. Accordingly, models of cuscuton
gravity are shown to possess non-singular cosmological solutions and those
appear stable at first sight. Yet, various subtleties arise in the
perturbations such as apparent divergences, e.g., when the Hubble parameter
crosses zero. We revisit the cosmological perturbations in various gauges and
demonstrate that the stability results are robust even at those crossing
points, although certain gauges are better suited to analyze the perturbations.
In particular, the spatially-flat gauge is found to be ill defined when $H=0$.
Otherwise, the sound speed is confirmed to be generally close to unity in the
ultraviolet, and curvature perturbations are shown to remain essentially
constant in the infrared throughout a bounce phase. Perturbations for a model
of extended cuscuton (as a subclass of Horndeski theory) are also studied and
similar conclusions are recovered.
