The broad division of the old world into Europe, Asia and Africa, at least for authors of the Mediterranean world, was essentially established by at least the end of the sixth century BC by the Ionian geographers. Although, importantly, these are not 'continents' in a modern sense. That term doesn't begin to be applied to them until the 15th century. Rather, down through the Middle Ages, they are the tres partes (three parts) of the world. And while this distinction may seem a bit like splitting hairs, it highlights the conventional and culturally constructed nature of meta-geographical concepts like 'continents'. So there is no reason to think that any of these authors in particular conceived of the distinction as one between three autonomous land-masses (this sort of notion is a relatively modern development).

This is quite clear in the writing of Herodotus, by whose time these Europe, Africa and Asia had become a completely standard division of the world. Herodotus expresses his confusion at why a single landmass should be given three different names:

For my part I cannot conceive why three names, and women's names especially, should ever have been given to a tract which is in reality one, nor why the Egyptian Nile and the Colchian Phasis (or according to others the Maeotic Tanais and Cimmerian ferry) should have been fixed upon for the boundary lines; nor can I even say who gave the three tracts their names, or whence they took the epithets. According to the Greeks in general, Libya was so called after a certain Libya, a native woman, and Asia after the wife of Prometheus. The Lydians, however, put in a claim to the latter name, which, they declare, was not derived from Asia the wife of Prometheus, but from Asies, the son of Cotys, and grandson of Manes, who also gave name to the tribe Asias at Sardis. As for Europe, no one can say whether it is surrounded by the sea or not, neither is it known whence the name of Europe was derived, nor who gave it name, unless we say that Europe was so called after the Tyrian Europe, and before her time was nameless, like the other divisions. But it is certain that Europe was an Asiatic, and never even set foot on the land which the Greeks now call Europe, only sailing from Phoenicia to Crete, and from Crete to Lycia. However let us quit these matters. We shall ourselves continue to use the names which custom sanctions. (Histories 4.37)

But the actual boundaries of the continents is not altogether clear in the ancient world. We can see this already in this quotation, where Herodotus suggests that while others use the north-south running Tanais (normally identified as the Don River), he prefers the east-west running Phasis (which I believe was a hypothetical construct). As such, following the Ionians, Herodotus divides Europe and Asia east-west, making them the northern and southern halves of the oecumene (with Africa often relegated to a province of Asia).

Now the Greeks did imagine that there were certain geographical differences between these regions, but the connection between continents and climates was frequently hazy. The Ionians developed an account of the world's climates that was based around a north-south, cold-hot, continuum. This in part explains the draw of a division between a northern Europe and a southern Asia. However, the most consistent discussion of this theme that we find comes in the Hippocratic text, Airs, Waters, Places. The central section of the text begins with the declaration that:

I wish to show, respecting Asia and Europe, how, in all respects, they differ from one another, and concerning the figure of the inhabitants, for they are different, and do not at all resemble one another. To treat of all would be a long story, but I will tell you how I think it is with regard to the greatest and most marked differences. I say, then, that Asia differs very much from Europe as to the nature of all things....

The Airs, however, considers the four cardinal winds as the dominant influence over the parts of the world, a much more intricate and well developed system than the normal north-south climatic spectrum that dominated the majority of ancient and medieval thought. Nevertheless, it provides the most clear and consistent connection between the parts of the world and their having distinct climatic characteristics, which determine the physical characteristics of the inhabitants.

By the Hellenistic period, Herodotus's preference for the Phasis had wained and the Tanais had essentially won out as the canonical boundary between Europe and Asia. Nevertheless, this didn't come alongside a decisive shift towards an eastern Asia and a western Europe. Instead, we find consistent confusion, or at least mixed messages, about the proper way to divide the regions of the world. For example, Strabo, around the turn of the first century AD, tells a story about how Alexander the Great misidentified the Tanais as an east-west flowing river, connecting the Black sea and the Caspian sea:

Many false notions were also added to the account of this sea because of Alexander's love of glory; for, since it was agreed by all that the Tanaïs separated Asia from Europe, and that the region between the sea and the Tanaïs, being a considerable part of Asia, had not fallen under the power of the Macedonians, it was resolved to manipulate the account of Alexander's expedition so that in fame at least he might be credited with having conquered those parts of Asia too. They therefore united Lake Maeotis, which receives the Tanaïs, with the Caspian Sea, calling this too a lake and asserting that both were connected with one another by an underground passage and that each was a part of the other. Polycleitus goes on to adduce proofs in connection with his belief that the sea is a lake 510 (for instance, he says that it produces serpents, and that its water is sweetish); and that it is no other than Maeotis he judges from the fact that the Tanaïs empties into it. From the same Indian mountains, where the Ochus and the Oxus and several other rivers rise, flows also the Iaxartes, which, like those rivers, empties into the Caspian Sea and is the most northerly of them all. This river, accordingly, they named Tanaïs. (Geography 11.7.4)

Likewise, in the late second century BC, Varro has a character explain that the world is naturally divided into north and south, a division that he links to the division of Europe and Asia, and associates specifically with Eratosthenes:

When we had taken our seats Agrasius opened the conversation: "You have all travelled through many lands; have you seen any land more fully cultivated than Italy?" "For my part," replied Agrius, "I think there is none which is so wholly under cultivation. Consider first: Eratosthenes, following a most natural division, has divided the earth into two parts, one to the south and the other to the north; and since the northern part is undoubtedly more healthful than the southern, while the part which is more healthful is more fruitful, we must agree that Italy at least was more suited to cultivation than Asia. In the first place, it is in Europe; and in the next place, this part of Europe has a more temperate climate than we find farther inland. For the winter is almost continuous in the interior, and no wonder, since its lands lie between the arctic circle and the pole, where the sun is not visible for six months at a time; wherefore we are told that even navigation in the ocean is not possible in that region because of the frozen sea." (De re rustica 1.2)

Indeed, we find the same attribution in Strabo:

In the Third Book of his Geography Eratosthenes, in establishing the map of the inhabited world, divides it into two parts by a line drawn from west to east, parallel to the equatorial line; and as ends of this line he takes, on the west, the Pillars of Heracles, on the east, the capes and most remote peaks of the mountain-chain that forms the northern boundary of India. (Geography 2.1.1)

It is not until the first century AD that, through with the increase Roman expansion to the east, the great eastern extension of Asia was really fully recognised. Along with this recognition comes the ultimate solidification of an eastern Asia and a western Europe. While the precise dynamics of this change are somewhat obscure, when Sallust, writing in the middle of the first century BC, gives a brief geography of the world in his Jugurthine War, unlike the preceding authors, he adjoins Africa to Europe, not to Asia, so as to properly establish a bipartition of the world east and west between Asia and Europe:

In their division of the earth's surface geographers commonly regard Africa as a third part, a few recognize only Asia and Europe, including Africa in the latter. (17.3)

From this point onwards, more or less every major Roman geographer adheres to this division of a massive eastern Asia, equivalent in size to the combined western Europe and Africa, divided by three bodies of water: the Tanais, Nile and Mediterranean. This is what we find in Strabo, Pomponius Mela and Lucan, all writing in and around the first century AD.