  Many theories of gravity admit formulations in different, conformally related
manifolds, known as the Jordan and Einstein conformal frames. Among them are
various scalar-tensor theories of gravity and high-order theories with the
Lagrangian $f(R)$ where $R$ is the scalar curvature and $f$ an arbitrary
function. It may happen that a singularity in the Einstein frame corresponds to
a regular surface S_trans in the Jordan frame, and the space-time is then
continued beyond this surface. This phenomenon is called a conformal
continuation (CC). We discuss the properties of vacuum static, spherically
symmetric configurations of arbitrary dimension in scalar-tensor and $f(R)$
theories of gravity and indicate necessary and sufficient conditions for the
existence of solutions admitting a CC. Two cases are distinguished, when
S_trans is an ordinary regular sphere and when it is a Killing horizon. Two
explicit examples of CCs are presented.
