  We investigate cusp singularities in f(R) gravity, especially for Starobinsky
and Hu-Sawicki dark energy models. We illustrate that, by using double-null
numerical simulations, a cusp singularity can be triggered by gravitational
collapses. This singularity can be cured by adding a quadratic term, but this
causes a Ricci scalar bump that can be observed by an observer outside the
event horizon. Comparing with cosmological parameters, it seems that it would
be difficult to see super-Planckian effects by astrophysical experiments. On
the other hand, at once there exists a cusp singularity, it can be a mechanism
to realize a horizon scale curvature singularity that can be interpreted by a
firewall.
