The perturbative expansion of general relativity breaks at the two-loop level: one produces a term in the effective action which is cubic in the Weyl tensor, as showed by Goroff and Sagnotti, which has a UV-divergent coefficient, and which has to be canceled by a counterterm. This introduces a new unknown finite coupling to the "no longer Einstein-Hilbert action", too. Because of that, one ultimately gets an infinite number of unknown couplings and loses predictivity.
We say that general relativity is perturbatively non-renormalizable.
This shows that there has to be new physics that determines all the unknown parameters of the low-energy physics (and, typically, adds qualitatively new phenomena at high energies, too). The spectrum of possibilities is given by the "landscape" of string/M-theory. All versions of the Minkowski or anti de Sitter or de Sitter space span a complicated set with many components. Some of these components are discrete; we call them the stabilized vacua. Some of them have residual parameters - the moduli - and those are very interesting mathematically (and they are usually calculable, and often have some unbroken supersymmetry) but they are unacceptable phenomenologically.
Only the stabilized vacua are realistic candidates for a theory of the real world. The unstabilized ones violate the equivalence principle, allow the fine-structure constant and similar constants to vary, and lead to new, unobserved long-range forces.
However, around each vacuum, it's still true that the amplitudes may be Taylor-expanded in any coupling constant that happens to be weak. The fact that only one value of a coupling constant - or a scalar field, if we're in the gravitational context - is the right one is seen as the existence of a potential for this scalar field. If we find ourselves away from the minimum (or extremum) of this potential, there will exist non-zero one-point functions for this scalar field that drive the Universe back to the minimum.
When the computation is done properly, the scattering amplitudes are analytic functions of the energy-momentum vectors "almost everywhere". This fact is guaranteed by the locality (or even approximately locality) of the physical phenomena in spacetime. So there can't be any delta-functions here - except for those that impose the conservation laws.
So it is indeed true that the Minkowski-like or anti-de-Sitter-like "empty space" solutions only exist for the right values of the couplings that minimize the potential; the physical spectrum of "empty space" states is strictly vanishing away from the right stabilized value of the moduli. But this fact shouldn't be viewed as a discontinuity in the underlying maths. Instead, you should imagine that for wrong values of the coupling constants, there exist "analogous" solutions which are not "empty space". Instead, in these solutions, the scalar fields oscillate around their preferred value for which the potential is minimized.
The states "don't disappear" as you move to wrong values of the coupling constant. Instead, they just fail to be translationally symmetric in time. In this sense, there's no discontinuity and the calculations of physical amplitudes and other observables never involve delta-functions of the scalar fields.
Much more generally, however, it is of course conceivable that the perturbative expansions break down for many known and unknown reasons. It's been known for a long time that the perturbative expansions ultimately diverge; and they don't capture all the physical phenomena, anyway. If we resum the terms up to the minimal one (before they start to blow up again) in a divergent expansion in a coupling constant, the minimum term - an uncertainty of the sum - is of the same order as the first non-perturbative contributions to the amplitudes (various instantons). But attempts to "resum the perturbative expansions more properly" are not the only methods to approach the non-perturbative physics.
On the other hand, if a perturbative expansion were "qualitatively" failing even for a very small value of the coupling constant, it would probably prove that the theory is inconsistent. This shouldn't happen. Even if it happened in a hypothetical theory, you would have to think how this theory allows to calculate something, at least in principle - without that, you shouldn't talk about a theory at all.