  We study the structure of a family of static, spherically symmetric
space-times generated by an anisotropic fluid and governed by a particular type
of $f(R)$ theory. We find that for a range of parameters with physical
interest, such solutions represent black holes with the central singularity
replaced by a finite size wormhole. We show that time-like geodesics and null
geodesics with nonzero angular momentum never reach the wormhole throat due to
an infinite potential barrier. For null radial geodesics, it takes an infinite
affine time to reach the wormhole. This means that the resulting space-time is
geodesically complete and, therefore, nonsingular despite the generic existence
of curvature divergences at the wormhole throat.
