  The junction conditions for the infinite derivative gravity theory
${R{+}RF(\Box)R}$ are derived under the assumption that the conditions can be
imposed by avoiding the `ill-defined expressions' in the theory of
distributions term by term in infinite summations. We find that the junction
conditions of such non-local theories are much more restrictive than in local
theories, since the conditions comprise an infinite number of equations for the
Ricci scalar. These conditions can constrain the geometry far beyond the
matching hypersurface. Furthermore, we derive the junction field equations
which are satisfied by the energy-momentum on the hypersurface. It turns out
that the theory still allows some matter content on the hypersurface (without
external flux and external tension), but with a traceless energy-momentum
tensor. We also discuss the proper matching condition where no matter is
concentrated on the hypersurface. Finally, we explore the possible applications
and consequences of our results to the braneworld scenarios and star models.
Particularly, we find that the internal tension is given purely by the trace of
the energy-momentum tensor of the matter confined to the brane. Consequences of
the junction conditions are illustrated on two simple examples of static and
collapsing stars. It is demonstrated that even without solving the field
equations the geometry on one side of the hypersurface can be determined to a
great extent by the geometry on the other side if the Ricci scalar is analytic.
We further show that some usual star models in the general relativity are no
longer solutions of the infinite derivative gravity.
