I am afraid Demetrius your sources are quite inaccurate.We cannot talk about 'some' continuity,because there was never a gap in talking Greek.The language changed because the needs of the people changed,and that's because language is a tool not a heirloom.The language is alive only when it serves the needs of the people.The fact that Greek adapted to the changes of culture and life is proof of its healthy evolution.What the modern Greeks tried to introduce was ancient Greek as the Hellenistic writers tried to reintroduce attic Greek (there were many such movements even in the Byzantine era), not reinvent Greek;that is preposterous to talk or even assume and proves a misleading luck of knowledge.The 'kathareuousa' indeed was artificial, but scholars do not care about the formers link to ancient Greek but about the popular tongue to the latter.It was the popular tongue and its many diallects that proved the unbreakable continuity with the ancient modes of the language, eg Cypriotic and Pontic diallects.The case of the evolution of Greek and its continuity cannot be compared to the evolution of English or German that were altogether modified.Words were not reinvented and a great bulk of the vocabulary was in continuous use'these words are not ancient, medieval or modern, they are Greek.Words like oikia, filos, stratiotes,ekklesia,demos,istoria never fell out of use!The modern Greeks are closer to ancient Greek not due to the Kathareuousa,but because before the introduction and establishment of kathareuousa they were in contact with older literature and recognised the continuity.In the 14th- 15th century(a time when most of the radical changes in Greek were already made) Gemistos Plethon, a byzantine scholar,said emphaticaly:'we are Greeks as our tongue and culture proves us to be'.Later on Adamantios Koraes, a Greek scholar of the 18th century, supporter of the popular tongue and against the artificial reinvention of Ancient Greek,said commenting on the former mode of the language:'if Plato and Aristotle and all the great of antiquity were to come back it would take them only a month to understand what we are speaking now'.It is an illusion to expect that the language would not undergo changes'that would be suspicious as it would try to persuade us that no changes took place, but change is the law of the universe. As Heracleitus would have put it, everything changes to be the same again,and in fundamental aspects the structure of Greek and the vocabulary survived,i.e. the rules of creating the language.Greek is not static as other tongues;it is a 'spinal' language:u can go creating as many words and expresions as u want to suit your own particular purposes if u know how to use it.It is a highly individualistic language that provides a lot of freedom for idiolects to advance (unlike English that enjoys homogeinity).That is the flexibility that makes it stronger than other languages.Proof of this is the fact that language is a reflection of mentality.Now, Germanic and English and mostly European mentality have a flare for homogeneous things, unlike Greek that embraces diversity.It was this language that enabled the grasp of higher concepts and of counter-arguments to a degree of profoundness and sophistication.
There is a certain feeling of security in launching Greeks into oblivion.It is more comfortable to think of them as an idyllic utopia that is safely in the arms of the past against which we don't have to compete anymore.We have a flare for being nostalgic of lost civilisations.But if this is the case than I suggest we should look at another civilisation like the Hittites, because this is not the case for the Greeks who have been around always continuously shifting their cultural cloaks.But this was the case with antiquity as well;we cannot say there is one clear-cut Hellenic mode, vast are the differences between Myceanean Hellenism to Archaic and that to Classic and that to Hellenistic (where everything changed)or Graeco-Roman;but at the end of the day it is all Greek.If we are to say that modern Greek has 'some' continuity to ancient Greek we should be prepared to apply the same predicate to Hellenistic Greek when the most radical changes took place!
If one wants to decide on the continuity of Greek language and culture I suggest he learns modern Greek (thus a lot of concepts Engilish misunderstands or cannot conceive will be rendered explicit,as the structure of suntactic and grammatic thought-pattern is basically the same:a 'spinal' one)and have a look at the literature and culture of the mediaeval period (which never was a 'dark age' for the Greek Orient)and the bulk of modern Greek literature which in a span of two centuries is larger than that of antiquity.Let anyone take a look at the Greek folk songs of the 15th century ,which are a popular creation, not influenced by the Church or the Scholars, and let me know if he does not recognise Greek in them.Their proof as a link to ancient culture transcends languages and shows a continuity in the themes and patterns of the poems.It has been found that the Homeric Epics and the modern folk songs use almost the same expressions to convey an image or a meaning-something that led scholars to believe that the Homeric Epics were nothing more than an ingenius modification of Myceanean folk-songs.These themes survived among the mere uneducated people for centuries.And though I study classics the quality of modern literature is second to none.I can only dare any to read Kavafis, Palamas, Seferis, Elytis, Kazantzakis and numerous others.And of course we shouldn't forget the heroic stand of the 'Few' against the Nazi in the 2nd WW, when they were assimilated (by the foreign press)to the heroic stand of Thermopylae.The contribution of Greece to the Cause against fashism says a lot on the continuation of ideals by modern Greeks , which like the words 'eleutheria'(freedom) and 'andreia'(bravery)are not ancient or modern, they are Greek!