  We study two exactly solvable five-dimensional thick brane world models in
pure metric $f(R)$ gravity. Working in the Einstein frame, we show that these
solutions are stable against small linear perturbations, including the tensor,
vector, and scalar modes. For both models, the corresponding gravitational zero
mode is localized on the brane, which leads to the four-dimensional Newton's
law; while the massive modes are nonlocalized and only contribute a small
correction to the Newton's law at a large distance.
