  Several features of an $f(R)$ theory in which there is a maximum value for
the curvature are analyzed. The theory admits the vaccuum solutions of GR, and
also the radiation evolution for the scale factor of the standard cosmological
model. Working in the Jordan frame, a complete analysis of the phase space is
performed, and its results supported with examples obtainted by numerical
integration. In particular, we showed that theory has nonsingular cosmological
solutions which after the bounce enter a phase of de Sitter expansion and
subsequently relax to a GR-like radiation-dominated evolution.
