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

Pure geometric thick $f(R)$-branes: stability and localization of gravity