  We perform a careful investigation of the problem of physically realistic
gravitational collapse of massive stars in f(R)-gravity. We show that the extra
matching conditions that arise in the modified gravity imposes strong
constraints on the stellar structure and thermodynamic properties. In our
opinion these constraints are unphysical. We prove that no homogeneous stars
with non-constant Ricci scalar can be matched smoothly with a static exterior
for any nonlinear function f(R). Therefore, these extra constraints make
classes of physically realistic collapse scenarios in general relativity,
non-admissible in these theories. We also find an exact solution for an
inhomogeneous collapsing star in the Starobinski model that obeys all the
energy and matching conditions. However, we argue that such solutions are
fine-tuned and unstable to matter perturbations. Possible consequences on black
hole physics and the cosmic censorship conjecture are also discussed.
