  Spacetime singularity has always been of interest since the proof of the
Penrose-Hawking singularity theorem. Naked singularity naturally emerges from
reasonable initial conditions in the collapsing process. A recent interesting
approach in black hole information problem implies that we need a firewall to
break the surplus entanglements among the Hawking photons. Classically, the
firewall becomes a naked singularity. We find some vacuum analytical solutions
in $R^n$-gravity of the firewall-type and use these solutions as concrete
models to study the naked singularities. By using standard quantum theory, we
investigate the Hawking radiation emitted from the black holes with naked
singularities. Here we show that the singularity itself does not destroy
information. A unitary quantum theory works well around a firewall-type
singularity. We discuss the validity of our result in general relativity.
Further our result demonstrates that the temperature of the Hawking radiation
still can be expressed in the form of the surface gravity divided by $2\pi$.
This indicates that a naked singularity may not compromise the Hakwing
evaporation process.
