  We investigate which Jordan frame $F(R)$ gravity can describe a Type IV
singular bouncing cosmological evolution, with special emphasis given near the
point at which the Type IV singularity occurs. The cosmological bounce is
chosen in such a way so that the bouncing point coincides exactly with Type IV
singularity point. The stability of the resulting $F(R)$ gravity is examined
and in addition, we study the Einstein frame scalar-tensor theory counterpart
of the resulting Jordan frame $F(R)$ gravity. Also, by assuming that the Jordan
frame metric is chosen in such a way so that, when conformally transformed in
the Einstein frame, it yields a quasi de Sitter or de Sitter
Friedmann-Robertson-Walker metric, we study the observational indexes which
turn out to be consistent with Planck 2015 data in the case of the Einstein
frame scalar theory. Finally, we study the behavior of the effective equation
of state corresponding to the Type IV singular bounce and after we compare the
resulting picture with other bouncing cosmologies, we critically discuss the
implications of our analysis.
