  We consider static, spherically symmetric vacuum solutions to the equations
of a theory of gravity with the Lagrangian f(R) where R is the scalar curvature
and f is an arbitrary function. Using a well-known conformal transformation,
the equations of f(R) theory are reduced to the ``Einstein picture'', i.e., to
the equations of general relativity with a source in the form of a scalar field
with a potential. We have obtained necessary and sufficient conditions for the
existence of solutions admitting conformal continuations. The latter means that
a central singularity that exists in the Einstein picture is mapped, in the
Jordan picture (i.e., in the manifold corresponding to the original formulation
of the theory), to a certain regular sphere, and the solution to the field
equations may be smoothly continued beyond it. The value of the curvature R on
the transition sphere corresponds to an extremum of the function f(R). Specific
examples are considered.
