  The existence of black holes is a central prediction of general relativity
and thus serves as a basic consistency test for modified theories of gravity.
In spherical symmetry, only two classes of dynamic solutions are compatible
with the formation of an apparent horizon in finite time of a distant observer.
Moreover, the formation of black holes follows a unique scenario involving both
types of solutions. To be compatible with their existence, any self-consistent
theory of modified gravity must satisfy several constraints. We derive
properties of the modified gravity terms of f(R) and generic fourth-order
gravity theories and find that they naturally accommodate both classes of
solutions. Consequently, the observation of an apparent horizon by itself may
not suffice to distinguish between general relativity and modifications
including up to fourth-order derivatives in the metric.
