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It’s not only possible, it exists as you describe. Italian, French (Provençal, really), Catalan, Spanish and Portuguese (and many local dialects thereof) merge continuously into each other, and everyone can understand their near neighbours in both directions, even across international borders where the language nominally changes, but Sicilians and Portuguese can’t understand each other. National broadcast media are reducing this by standardising language within each country, so your world should have a lower technology level.

It's almost like Slavic languages, though the reason has more to do with history of literacy, political changes and assimilation.

Slovak is mutually intelligible with modern heavily Slovakized Rusyn, which is mutually intelligible with Rusyn as spoken in Transcarpathia, which is almost mutually intelligible with standard Ukrainian, which is mutually intelligible with Surzhik, which is mutually intelligible with Russian, which is mutually intelligible with Belarussian, which is not really, but somewhat mutually intelligible with Polish, which is almost mutually intelligible with Góral language (at the Polish side of the border), which is basically the same as the Góral language at the Slovak side of the border, which is kind of mutually intelligible with standard Slovak.

Apart from the broken link between Belarussian and Polish (which in an alternative history of surviving Polish-Lithuanian Commonwealth could exist), there is almost a complete ring.

Going the southern route, central Slovak has some tenuous connection with Croatian (though mutual intelligibility has been already lost to history); Croatian blends into other (neo-štokavian) Balkan languages, transitions into Macedonian, which is mutually intelligible with Bulgarian, which is surprisingly close in lexica to Russian, because of the heavy influence of Old Church Slavonic on Russian. From Russian, the usual route via Surzhik->Ukrainian->Transcarpathian Russyn->Slovak Russyn->Slovak closes the ring.

There is the missing step from Croatian to Slovak, and the connection between Bulgarian and Russian is rather weak. But still.

Old Norse and Old English

It's basically how languages spread in pre-modern times. Have you seen "Vikings"? There is a scene in the first season when Ragnar lands in England and meets a local landlord and his soldiers on the beach. They're not able to communicate immediately. The vikings speak Old Norse, the English - well, Old English. It's exactly the situation you described: these are both languages coming form the common proto-Germanic roots, but they are separated by hundreds or maybe a thousand years of development. The only difference from what you wrote is that no language stays the same - the "original" language of people who were left behind in Scandinavia also evolved in time.

I wrote an article about it for a Polish linguistic website woofla.pl, but it's in Polish. All I have in English is an animation showing the development of Germanic languages: https://www.youtube.com/watch?v=UzYjsuvHx4c

In short, Germanic people settled in about 2000-1500BCE in the area of the modern Denmark and southern Sweden, and lived there in relative isolation until around 500-200BCE. The isolation contributed to the differences between Germanic and all other Indo-European families of languages. At the end of that period a dialectal continuum emerged with people who dwelt in southern and western parts on one end of the spectrum, and population in the Scandinavia on the other. From the first group, Western Germanic languages emerged, from the second, Old Norse.

Western Germanic languages were also a continuum for a long time (even until the modern times to be precise - even in the 18th century the border between Dutch and Low German was kinda fuzzy). In the south, close to the Alps, Old High German started to develop. In the north, Low German dialects. One of them was Saxon which was probably close or the same as the languages of Angles, and together these two peoples invaded Britain in the 5th-6th century CE. Old English developed as a merger of Old Saxon, Old Frisian, and some other, smaller dialects of people who joined the invaders. Some of them might have even spoken the northern dialects.

In the same time Old Norse was developing as well, but in a more conservative fashion. In the times of Ragnar, late 8th and early 9th century, it was still one language with just some differences between east and west Scandinavia. Even now Danish, Swedish and Norwegian (which has its own interesting story and is divided into two main "versions", Bokmal and Nynorsk) are quite intelligible.

Anyway, when the vikings invaded England it was as if the linguistic history made a full circle. And now both peoples needed translators. But not for long: the languages were still similar enough that probably after a few years of learning a Dane was able to speak Old English quite well, and vice versa. Only the later Norman invasion introduced a lot of (Norman) French vocabulary and put English on a very different track.

I think that what you are asking is more or less what happened with Latin and neo-Latin languages.

Take Romanian, French and Italian. They are all from Latin origin: an Italian can understand some French and some Romanian (actually Romanian sounds really similar to some South Italian dialects, while some Northern Italy dialects sound really close to French), but I doubt a Romanian and a French would had the same level of partial understanding.

This situation exists in Japan. While the dialect of Tokyo has become the standard language via radio and television, each region also has local dialects. As described, occupants of neighboring regions can understand one another, but travel any significant distance, and the local dialects become mutually incomprehensible. I recall a conversation with a man who was about 7 years old when evacuee children from Tokyo arrived and lived in local temples. He said they could not understand one another at all. He also said that he preferred to speak the local dialect; that he felt more himself while doing so.

Czech-Polish dialect continuum

(Wikimedia Commons, GDFL)

Czech and Polish are, for most native speakers of both languages, not mutually intelligible. However, in the easternmost corner of the Czech Republic, there is a transitional Cieszyn Silesian dialect (G2 in the map above). It is mutually intelligible with the Silesian dialect spoken on the Polish side of the border (G1) and, to a lesser extent, also with the Lachian dialects (C4) on the Czech side. So, there seems to be a well established dialect continuum in this region.

However, this continuum does not exist all along the Czech-Polish border. In fact, there are basically no transitional dialects outside the Cieszyn region.

The thing is, most of what is now the Czech-Polish border region used to be a predominantly German speaking area. After the expulsion of Germans from Czechoslovakia, new settlers from all over the country arrived to the region, bringing their own dialects, which eventually merged into what is now described as the mixed dialects (C5). A similar process took place on the Polish side where the so called Recovered Territories were settled from central and eastern Poland, again creating mixed dialects (B4). These new dialects of Czech and Polish are about as far apart as the respective standard languages are. So, no cross-border mutual intelligibility. Not without a significant exposure to the other language, anyway.

Definitely possible, although I don't know real examples from our world.

The Slavonic languages on the Balkans form a somewhat broken ring, almost encircling the Hungarian and Romanian language area: Going counterclockwise from Ukrainian via Ruthenian and Slovak to (first gap in the ring) Slovene, Croatian, Serbian, Macedonian and Bulgarian there is a dialect continuum. Between Bulgarian and Ukrainian is the second gap in the ring, when it were closed along the Black sea coast, Bulgarian and Ukrainian would exhibit a hard language boundary.

I would like to add another comparison in a lateral direction - time. This is exactly how language development over time works. Grandparents and their grandchildren can communicate, but if you were to move through a few generations, you wouldn't be able to understand your ancestors anymore and vice versa, even though every 'temporal subset' was able to understand each other.

More examples for you to use for research, since everyone has answered this well: Quebec French, Creole French, and French are all completely distinct.

In China, there are over 104 (?) dialects. They are distinct enough that not even neighbors can understand each other, so they communicate in English instead.

In Japan, the main island has it's form of Japanese, with a dialect in the Osaka prefecture being like American Southern English, as well as the Ryukyuan Islands having distinct languages that are Japonic, however are also unintelligible to Japanese-speakers.

Not only can it exist, it does, with Quebecois to Acadian/Chiac to true French/Parisenne

People from Paris cannot (easily) converse with those from Quebec as the dialects are quite different. People from Atlantic Canada (Acadians) can converse with people from Quebec, or from Paris.

Historically, Acadians and Quebecois were both from France; almost 400 years ago. The language in France has remained very much unchanged due to the Académie française. The language in Quebec evolved naturally.

In Atlantic Canada, though we speak a different dialect (a lot of unique words, and a lot of English*) we're taught Parisian French and expected to speak it in school. As Quebec is also part of Canada, and is the only province between Atlantic Canada and the rest of Canada, the dialects are geographically and politically related.

Probably related, about 350 years ago Lousiana French diverged as well, due to Le Grand Dérangement. My understanding is that "Cajun" is a bastardization of "Acadian" and that the dialects are very similar. That being said I would doubt that French Education in the American South follows 'Académie guidelines.

• "J'ai allé a la wharf pour fishé en ma boat" is an example of a "French" sentence with English syntax and words introduced.

For such a language to exist, it is sufficient that the dialects differ on at least two dimensions (principal components). Given the complexity of language, this is obviously possible. For example language L1 could share set of words S1 with L2. L2 shares S2 with L2. L3 shares S3 with L1. S1, S2 and S3 are all disjoint.

This sort of dialectal ring continuum might happen naturally for a population with homogeneous language origins living on a contiguously traversable, toroidal world with very limited travel distances. Essentially, a people speaking one language landed as one group on a donut without any great oceans separating its land masses. You might also be able to pull this off on a world which can only be traversed or inhabited along its equatorial zone, so long as it remains reasonably contiguous.

The more you increase the distance the average person travels, the more you dilute the dialects. To get the most varied continuum, reduce the average person to a walking distance (cf. "God Emperor of Dune").

Speakers of neighboring dialects will tend to retain mutually intelligibility and, since no one lives in a vacuum, there will be some "cross-pollination" of dialects across the spectrum. As these people spread across the face of this world, the dialects will diverge.

Eventually, the population will expand until one extreme meets the other. The first encounter would likely be between mutually unintelligible dialects. Whether they initially meet with war or trade, those two groups will gradually reach a point of relative stasis and a pidgin will emerge. This will develop into its own dialect and start propagating along the continuum. Of course, so long as the average person doesn't travel very far, these propagations will rapidly diminish with distance.